def local_coordinates_at_nonweierstrass(self, P, prec=20, name='t'):
"""
For a non-Weierstrass point `P = (a,b)` on the hyperelliptic
curve `y^2 = f(x)`, return `(x(t), y(t))` such that `(y(t))^2 = f(x(t))`,
where `t = x - a` is the local parameter.
INPUT:
- ``P = (a, b)`` -- a non-Weierstrass point on self
- ``prec`` -- desired precision of the local coordinates
- ``name`` -- gen of the power series ring (default: ``t``)
OUTPUT:
`(x(t),y(t))` such that `y(t)^2 = f(x(t))` and `t = x - a`
is the local parameter at `P`
EXAMPLES::
sage: R.<x> = QQ['x']
sage: H = HyperellipticCurve(x^5-23*x^3+18*x^2+40*x)
sage: P = H(1,6)
sage: x,y = H.local_coordinates_at_nonweierstrass(P,prec=5)
sage: x
1 + t + O(t^5)
sage: y
6 + t - 7/2*t^2 - 1/2*t^3 - 25/48*t^4 + O(t^5)
sage: Q = H(-2,12)
sage: x,y = H.local_coordinates_at_nonweierstrass(Q,prec=5)
sage: x
-2 + t + O(t^5)
sage: y
12 - 19/2*t - 19/32*t^2 + 61/256*t^3 - 5965/24576*t^4 + O(t^5)
AUTHOR:
- Jennifer Balakrishnan (2007-12)
"""
d = P[1]
if d == 0:
raise TypeError(
"P = %s is a Weierstrass point. Use local_coordinates_at_weierstrass instead!"
% P)
pol = self.hyperelliptic_polynomials()[0]
L = PowerSeriesRing(self.base_ring(), name)
t = L.gen()
L.set_default_prec(prec)
K = PowerSeriesRing(L, 'x')
pol = K(pol)
x = K.gen()
b = P[0]
f = pol(t + b)
for i in range((RR(log(prec) / log(2))).ceil()):
d = (d + f / d) / 2
return t + b + O(t**(prec)), d + O(t**(prec))
def local_coordinates_at_nonweierstrass(self, P, prec=20, name='t'):
"""
For a non-Weierstrass point `P = (a,b)` on the hyperelliptic
curve `y^2 = f(x)`, return `(x(t), y(t))` such that `(y(t))^2 = f(x(t))`,
where `t = x - a` is the local parameter.
INPUT:
- ``P = (a, b)`` -- a non-Weierstrass point on self
- ``prec`` -- desired precision of the local coordinates
- ``name`` -- gen of the power series ring (default: ``t``)
OUTPUT:
`(x(t),y(t))` such that `y(t)^2 = f(x(t))` and `t = x - a`
is the local parameter at `P`
EXAMPLES::
sage: R.<x> = QQ['x']
sage: H = HyperellipticCurve(x^5-23*x^3+18*x^2+40*x)
sage: P = H(1,6)
sage: x,y = H.local_coordinates_at_nonweierstrass(P,prec=5)
sage: x
1 + t + O(t^5)
sage: y
6 + t - 7/2*t^2 - 1/2*t^3 - 25/48*t^4 + O(t^5)
sage: Q = H(-2,12)
sage: x,y = H.local_coordinates_at_nonweierstrass(Q,prec=5)
sage: x
-2 + t + O(t^5)
sage: y
12 - 19/2*t - 19/32*t^2 + 61/256*t^3 - 5965/24576*t^4 + O(t^5)
AUTHOR:
- Jennifer Balakrishnan (2007-12)
"""
d = P[1]
if d == 0:
raise TypeError("P = %s is a Weierstrass point. Use local_coordinates_at_weierstrass instead!"%P)
pol = self.hyperelliptic_polynomials()[0]
L = PowerSeriesRing(self.base_ring(), name)
t = L.gen()
L.set_default_prec(prec)
K = PowerSeriesRing(L, 'x')
pol = K(pol)
x = K.gen()
b = P[0]
f = pol(t+b)
for i in range((RR(log(prec)/log(2))).ceil()):
d = (d + f/d)/2
return t+b+O(t**(prec)), d + O(t**(prec))
def local_coordinates_at_weierstrass(self, P, prec=20, name='t'):
"""
For a finite Weierstrass point on the hyperelliptic
curve y^2 = f(x), returns (x(t), y(t)) such that
(y(t))^2 = f(x(t)), where t = y is the local parameter.
INPUT:
- P a finite Weierstrass point on self
- prec: desired precision of the local coordinates
- name: gen of the power series ring (default: 't')
OUTPUT:
(x(t),y(t)) such that y(t)^2 = f(x(t)) and t = y
is the local parameter at P
EXAMPLES:
sage: R.<x> = QQ['x']
sage: H = HyperellipticCurve(x^5-23*x^3+18*x^2+40*x)
sage: A = H(4,0)
sage: x,y = H.local_coordinates_at_weierstrass(A,prec =5)
sage: x
4 + 1/360*t^2 - 191/23328000*t^4 + 7579/188956800000*t^6 + O(t^7)
sage: y
t + O(t^7)
sage: B = H(-5,0)
sage: x,y = H.local_coordinates_at_weierstrass(B,prec = 5)
sage: x
-5 + 1/1260*t^2 + 887/2000376000*t^4 + 643759/1587898468800000*t^6 + O(t^7)
sage: y
t + O(t^7)
AUTHOR:
- Jennifer Balakrishnan (2007-12)
"""
if P[1] != 0:
raise TypeError, "P = %s is not a finite Weierstrass point. Use local_coordinates_at_nonweierstrass instead!" % P
pol = self.hyperelliptic_polynomials()[0]
L = PowerSeriesRing(self.base_ring(), name)
t = L.gen()
L.set_default_prec(prec + 2)
K = PowerSeriesRing(L, 'x')
pol = K(pol)
x = K.gen()
b = P[0]
g = pol / (x - b)
c = b + 1 / g(b) * t**2
f = pol - t**2
fprime = f.derivative()
for i in range((RR(log(prec + 2) / log(2))).ceil()):
c = c - f(c) / fprime(c)
return c + O(t**(prec + 2)), t + O(t**(prec + 2))
def local_coordinates_at_weierstrass(self, P, prec=20, name="t"):
"""
For a finite Weierstrass point on the hyperelliptic
curve y^2 = f(x), returns (x(t), y(t)) such that
(y(t))^2 = f(x(t)), where t = y is the local parameter.
INPUT:
- P a finite Weierstrass point on self
- prec: desired precision of the local coordinates
- name: gen of the power series ring (default: 't')
OUTPUT:
(x(t),y(t)) such that y(t)^2 = f(x(t)) and t = y
is the local parameter at P
EXAMPLES:
sage: R.<x> = QQ['x']
sage: H = HyperellipticCurve(x^5-23*x^3+18*x^2+40*x)
sage: A = H(4,0)
sage: x,y = H.local_coordinates_at_weierstrass(A,prec =5)
sage: x
4 + 1/360*t^2 - 191/23328000*t^4 + 7579/188956800000*t^6 + O(t^7)
sage: y
t + O(t^7)
sage: B = H(-5,0)
sage: x,y = H.local_coordinates_at_weierstrass(B,prec = 5)
sage: x
-5 + 1/1260*t^2 + 887/2000376000*t^4 + 643759/1587898468800000*t^6 + O(t^7)
sage: y
t + O(t^7)
AUTHOR:
- Jennifer Balakrishnan (2007-12)
"""
if P[1] != 0:
raise TypeError, "P = %s is not a finite Weierstrass point. Use local_coordinates_at_nonweierstrass instead!" % P
pol = self.hyperelliptic_polynomials()[0]
L = PowerSeriesRing(self.base_ring(), name)
t = L.gen()
L.set_default_prec(prec + 2)
K = PowerSeriesRing(L, "x")
pol = K(pol)
x = K.gen()
b = P[0]
g = pol / (x - b)
c = b + 1 / g(b) * t ** 2
f = pol - t ** 2
fprime = f.derivative()
for i in range((RR(log(prec + 2) / log(2))).ceil()):
c = c - f(c) / fprime(c)
return c + O(t ** (prec + 2)), t + O(t ** (prec + 2))
def get_basic_integral(G, cocycle, gamma, center, j, prec=None):
p = G.p
HOC = cocycle.parent()
V = HOC.coefficient_module()
if prec is None:
prec = V.precision_cap()
Cp = Qp(p, prec)
verbose('precision = %s' % prec)
R = PolynomialRing(Cp, names='t')
PS = PowerSeriesRing(Cp, names='z')
t = R.gen()
z = PS.gen()
if prec is None:
prec = V.precision_cap()
try:
coeff_depth = V.precision_cap()
except AttributeError:
coeff_depth = V.coefficient_module().precision_cap()
resadd = ZZ(0)
edgelist = G.get_covering(1)[1:]
for rev, h in edgelist:
a, b, c, d = [Cp(o) for o in G.embed(h, prec).list()]
try:
c0val = 0
pol = PS(d * z + b) / PS(c * z + a)
pol -= Cp.teichmuller(center)
pol = pol**j
pol = pol.polynomial()
newgamma = G.Gpn(
G.reduce_in_amalgam(h * gamma.quaternion_rep,
return_word=False))
if rev: # DEBUG
newgamma = newgamma.conjugate_by(G.wp())
print 'reversing'
if G.use_shapiro():
mu_e = cocycle.evaluate_and_identity(newgamma)
else:
mu_e = cocycle.evaluate(newgamma)
if newgamma.quaternion_rep != 1:
print 'newgamma = ', newgamma
except AttributeError:
verbose('...')
continue
if HOC._use_ps_dists:
tmp = sum(a * mu_e.moment(i)
for a, i in izip(pol.coefficients(), pol.exponents())
if i < len(mu_e._moments))
else:
tmp = mu_e.evaluate_at_poly(pol, Cp, coeff_depth)
resadd += tmp
try:
if G.use_shapiro():
tmp = cocycle.get_liftee().evaluate_and_identity(newgamma)
else:
tmp = cocycle.get_liftee().evaluate(newgamma)
except IndexError:
pass
return resadd
def _e_bounds(self, n, prec):
p = self._p
prec = max(2,prec)
R = PowerSeriesRing(ZZ,'T',prec+1)
T = R(R.gen(),prec +1)
w = (1+T)**(p**n) - 1
return [infinity] + [valuation(w[j],p) for j in range(1,min(w.degree()+1,prec))]
def local_coordinates_at_weierstrass(self, P, prec=20, name='t'):
"""
For a finite Weierstrass point on the hyperelliptic
curve y^2 = f(x), returns (x(t), y(t)) such that
(y(t))^2 = f(x(t)), where t = y is the local parameter.
INPUT:
- P a finite Weierstrass point on self
- prec: desired precision of the local coordinates
- name: gen of the power series ring (default: 't')
OUTPUT:
(x(t),y(t)) such that y(t)^2 = f(x(t)) and t = y
is the local parameter at P
EXAMPLES:
sage: R.<x> = QQ['x']
sage: H = HyperellipticCurve(x^5-23*x^3+18*x^2+40*x)
sage: A = H(4, 0)
sage: x, y = H.local_coordinates_at_weierstrass(A, prec=7)
sage: x
4 + 1/360*t^2 - 191/23328000*t^4 + 7579/188956800000*t^6 + O(t^7)
sage: y
t + O(t^7)
sage: B = H(-5, 0)
sage: x, y = H.local_coordinates_at_weierstrass(B, prec=5)
sage: x
-5 + 1/1260*t^2 + 887/2000376000*t^4 + O(t^5)
sage: y
t + O(t^5)
AUTHOR:
- Jennifer Balakrishnan (2007-12)
- Francis Clarke (2012-08-26)
"""
if P[1] != 0:
raise TypeError(
"P = %s is not a finite Weierstrass point. Use local_coordinates_at_nonweierstrass instead!"
% P)
L = PowerSeriesRing(self.base_ring(), name)
t = L.gen()
pol = self.hyperelliptic_polynomials()[0]
pol_prime = pol.derivative()
b = P[0]
t2 = t**2
c = b + t2 / pol_prime(b)
c = c.add_bigoh(prec)
for _ in range(1 + log(prec, 2)):
c -= (pol(c) - t2) / pol_prime(c)
return (c, t.add_bigoh(prec))
def zeta_series(self, n, t):
"""
Return the zeta series.
Compute a power series approximation to the zeta function of a
scheme over a finite field.
INPUT:
- ``n`` -- the number of terms of the power series to
compute
- ``t`` -- the variable which the series should be
returned
OUTPUT:
A power series approximating the zeta function of self
EXAMPLES::
sage: P.<x> = PolynomialRing(GF(3))
sage: C = HyperellipticCurve(x^3+x^2+1)
sage: R.<t> = PowerSeriesRing(Integers())
sage: C.zeta_series(4,t)
1 + 6*t + 24*t^2 + 78*t^3 + 240*t^4 + O(t^5)
sage: (1+2*t+3*t^2)/(1-t)/(1-3*t) + O(t^5)
1 + 6*t + 24*t^2 + 78*t^3 + 240*t^4 + O(t^5)
Note that this function depends on count_points, which is only
defined for prime order fields for general schemes.
Nonetheless, since :trac:`15108` and :trac:`15148`, it supports
hyperelliptic curves over non-prime fields::
sage: C.base_extend(GF(9,'a')).zeta_series(4,t)
1 + 12*t + 120*t^2 + 1092*t^3 + 9840*t^4 + O(t^5)
"""
F = self.base_ring()
if not F.is_finite():
raise TypeError(
'zeta functions only defined for schemes over finite fields')
try:
a = self.count_points(n)
except AttributeError:
raise NotImplementedError(
'count_points() required but not implemented')
R = PowerSeriesRing(Rationals(), 'u')
u = R.gen()
temp = sum(a[i - 1] * (u.O(n + 1))**i / i for i in range(1, n + 1))
temp2 = temp.exp()
return (temp2(t).O(n + 1))
def local_coordinates_at_weierstrass(self, P, prec=20, name="t"):
"""
For a finite Weierstrass point on the hyperelliptic
curve y^2 = f(x), returns (x(t), y(t)) such that
(y(t))^2 = f(x(t)), where t = y is the local parameter.
INPUT:
- P a finite Weierstrass point on self
- prec: desired precision of the local coordinates
- name: gen of the power series ring (default: 't')
OUTPUT:
(x(t),y(t)) such that y(t)^2 = f(x(t)) and t = y
is the local parameter at P
EXAMPLES:
sage: R.<x> = QQ['x']
sage: H = HyperellipticCurve(x^5-23*x^3+18*x^2+40*x)
sage: A = H(4, 0)
sage: x, y = H.local_coordinates_at_weierstrass(A, prec=7)
sage: x
4 + 1/360*t^2 - 191/23328000*t^4 + 7579/188956800000*t^6 + O(t^7)
sage: y
t + O(t^7)
sage: B = H(-5, 0)
sage: x, y = H.local_coordinates_at_weierstrass(B, prec=5)
sage: x
-5 + 1/1260*t^2 + 887/2000376000*t^4 + O(t^5)
sage: y
t + O(t^5)
AUTHOR:
- Jennifer Balakrishnan (2007-12)
- Francis Clarke (2012-08-26)
"""
if P[1] != 0:
raise TypeError, "P = %s is not a finite Weierstrass point. Use local_coordinates_at_nonweierstrass instead!" % P
L = PowerSeriesRing(self.base_ring(), name)
t = L.gen()
pol = self.hyperelliptic_polynomials()[0]
pol_prime = pol.derivative()
b = P[0]
t2 = t ** 2
c = b + t2 / pol_prime(b)
c = c.add_bigoh(prec)
for _ in range(1 + log(prec, 2)):
c -= (pol(c) - t2) / pol_prime(c)
return (c, t.add_bigoh(prec))
def zeta_series(self, n, t):
"""
Return the zeta series.
Compute a power series approximation to the zeta function of a
scheme over a finite field.
INPUT:
- ``n`` -- the number of terms of the power series to
compute
- ``t`` -- the variable which the series should be
returned
OUTPUT:
A power series approximating the zeta function of self
EXAMPLES::
sage: P.<x> = PolynomialRing(GF(3))
sage: C = HyperellipticCurve(x^3+x^2+1)
sage: R.<t> = PowerSeriesRing(Integers())
sage: C.zeta_series(4,t)
1 + 6*t + 24*t^2 + 78*t^3 + 240*t^4 + O(t^5)
sage: (1+2*t+3*t^2)/(1-t)/(1-3*t) + O(t^5)
1 + 6*t + 24*t^2 + 78*t^3 + 240*t^4 + O(t^5)
Note that this function depends on count_points, which is only
defined for prime order fields for general schemes.
Nonetheless, since :trac:`15108` and :trac:`15148`, it supports
hyperelliptic curves over non-prime fields::
sage: C.base_extend(GF(9,'a')).zeta_series(4,t)
1 + 12*t + 120*t^2 + 1092*t^3 + 9840*t^4 + O(t^5)
"""
F = self.base_ring()
if not F.is_finite():
raise TypeError("zeta functions only defined for schemes over finite fields")
try:
a = self.count_points(n)
except AttributeError:
raise NotImplementedError("count_points() required but not implemented")
R = PowerSeriesRing(Rationals(), "u")
u = R.gen()
temp = sum(a[i - 1] * (u.O(n + 1)) ** i / i for i in range(1, n + 1))
temp2 = temp.exp()
return temp2(t).O(n + 1)
def zeta_series(self, n, t):
"""
Return the zeta series.
Compute a power series approximation to the zeta function of a
scheme over a finite field.
INPUT:
- ``n`` -- the number of terms of the power series to
compute
- ``t`` -- the variable which the series should be
returned
OUTPUT:
A power series approximating the zeta function of self
EXAMPLES::
sage: P.<x> = PolynomialRing(GF(3))
sage: C = HyperellipticCurve(x^3+x^2+1)
sage: R.<t> = PowerSeriesRing(Integers())
sage: C.zeta_series(4,t)
1 + 6*t + 24*t^2 + 78*t^3 + 240*t^4 + O(t^5)
sage: (1+2*t+3*t^2)/(1-t)/(1-3*t) + O(t^5)
1 + 6*t + 24*t^2 + 78*t^3 + 240*t^4 + O(t^5)
Note that this function depends on count_points, which is only
defined for prime order fields::
sage: C.base_extend(GF(9,'a')).zeta_series(4,t)
Traceback (most recent call last):
...
NotImplementedError: Point counting only implemented for schemes over prime fields
"""
F = self.base_ring()
if not F.is_finite():
raise TypeError('zeta functions only defined for schemes over finite fields')
try:
a = self.count_points(n)
except AttributeError:
raise NotImplementedError('count_points() required but not implemented')
R = PowerSeriesRing(Rationals(), 'u')
u = R.gen()
temp = sum(a[i-1]*(u.O(n+1))**i/i for i in range(1,n+1))
temp2 = temp.exp()
return(temp2(t).O(n+1))
def zeta_series(self, n, t):
"""
Return the zeta series.
Compute a power series approximation to the zeta function of a
scheme over a finite field.
INPUT:
- ``n`` - the number of terms of the power series to
compute
- ``t`` - the variable which the series should be
returned
OUTPUT:
A power series approximating the zeta function of self
EXAMPLES::
sage: P.<x> = PolynomialRing(GF(3))
sage: C = HyperellipticCurve(x^3+x^2+1)
sage: R.<t> = PowerSeriesRing(Integers())
sage: C.zeta_series(4,t)
1 + 6*t + 24*t^2 + 78*t^3 + 240*t^4 + O(t^5)
sage: (1+2*t+3*t^2)/(1-t)/(1-3*t) + O(t^5)
1 + 6*t + 24*t^2 + 78*t^3 + 240*t^4 + O(t^5)
Note that this function depends on count_points, which is only
defined for prime order fields::
sage: C.base_extend(GF(9,'a')).zeta_series(4,t)
Traceback (most recent call last):
...
NotImplementedError: Point counting only implemented for schemes over prime fields
"""
F = self.base_ring()
if not F.is_finite():
raise TypeError, "Zeta functions only defined for schemes over finite fields"
a = self.count_points(n)
R = PowerSeriesRing(Rationals(), 'u')
u = R.gen()
temp = sum(a[i - 1] * (u.O(n + 1))**i / i for i in range(1, n + 1))
temp2 = temp.exp()
return (temp2(t).O(n + 1))
class MFSeriesConstructor(SageObject,UniqueRepresentation):
r"""
Constructor for the Fourier expansion of some
(specific, basic) modular forms.
The constructor is used by forms elements in case
their Fourier expansion is needed or requested.
"""
@staticmethod
def __classcall__(cls, group = HeckeTriangleGroup(3), prec=ZZ(10)):
r"""
Return a (cached) instance with canonical parameters.
.. NOTE:
For each choice of group and precision the constructor is
cached (only) once. Further calculations with different
base rings and possibly numerical parameters are based on
the same cached instance.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.series_constructor import MFSeriesConstructor
sage: MFSeriesConstructor() == MFSeriesConstructor(3, 10)
True
sage: MFSeriesConstructor(group=4).hecke_n()
4
sage: MFSeriesConstructor(group=5, prec=12).prec()
12
"""
if (group==infinity):
group = HeckeTriangleGroup(infinity)
else:
try:
group = HeckeTriangleGroup(ZZ(group))
except TypeError:
group = HeckeTriangleGroup(group.n())
prec=ZZ(prec)
# We don't need this assumption the precision may in principle also be negative.
# if (prec<1):
# raise Exception("prec must be an Integer >=1")
return super(MFSeriesConstructor,cls).__classcall__(cls, group, prec)
def __init__(self, group, prec):
r"""
Constructor for the Fourier expansion of some
(specific, basic) modular forms.
INPUT:
- ``group`` -- A Hecke triangle group (default: HeckeTriangleGroup(3)).
- ``prec`` -- An integer (default: 10), the default precision used
in calculations in the LaurentSeriesRing or PowerSeriesRing.
OUTPUT:
The constructor for Fourier expansion with the specified settings.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.series_constructor import MFSeriesConstructor
sage: MFC = MFSeriesConstructor()
sage: MFC
Power series constructor for Hecke modular forms for n=3 with (basic series) precision 10
sage: MFC.group()
Hecke triangle group for n = 3
sage: MFC.prec()
10
sage: MFC._series_ring
Power Series Ring in q over Rational Field
sage: MFSeriesConstructor(group=4)
Power series constructor for Hecke modular forms for n=4 with (basic series) precision 10
sage: MFSeriesConstructor(group=5, prec=12)
Power series constructor for Hecke modular forms for n=5 with (basic series) precision 12
sage: MFSeriesConstructor(group=infinity)
Power series constructor for Hecke modular forms for n=+Infinity with (basic series) precision 10
"""
self._group = group
self._prec = prec
self._series_ring = PowerSeriesRing(QQ,'q',default_prec=self._prec)
def _repr_(self):
r"""
Return the string representation of ``self``.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.series_constructor import MFSeriesConstructor
sage: MFSeriesConstructor(group=4)
Power series constructor for Hecke modular forms for n=4 with (basic series) precision 10
sage: MFSeriesConstructor(group=5, prec=12)
Power series constructor for Hecke modular forms for n=5 with (basic series) precision 12
"""
return "Power series constructor for Hecke modular forms for n={} with (basic series) precision {}".\
format(self._group.n(), self._prec)
def group(self):
r"""
Return the (Hecke triangle) group of ``self``.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.series_constructor import MFSeriesConstructor
sage: MFSeriesConstructor(group=4).group()
Hecke triangle group for n = 4
"""
return self._group
def hecke_n(self):
r"""
Return the parameter ``n`` of the (Hecke triangle) group of ``self``.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.series_constructor import MFSeriesConstructor
sage: MFSeriesConstructor(group=4).hecke_n()
4
"""
return self._group.n()
def prec(self):
r"""
Return the used default precision for the PowerSeriesRing or LaurentSeriesRing.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.series_constructor import MFSeriesConstructor
sage: MFSeriesConstructor(group=5).prec()
10
sage: MFSeriesConstructor(group=5, prec=20).prec()
20
"""
return self._prec
@cached_method
def J_inv_ZZ(self):
r"""
Return the rational Fourier expansion of ``J_inv``,
where the parameter ``d`` is replaced by ``1``.
This is the main function used to determine all Fourier expansions!
.. NOTE:
The Fourier expansion of ``J_inv`` for ``d!=1``
is given by ``J_inv_ZZ(q/d)``.
.. TODO:
The functions that are used in this implementation are
products of hypergeometric series with other, elementary,
functions. Implement them and clean up this representation.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.series_constructor import MFSeriesConstructor
sage: MFSeriesConstructor(prec=3).J_inv_ZZ()
q^-1 + 31/72 + 1823/27648*q + O(q^2)
sage: MFSeriesConstructor(group=5, prec=3).J_inv_ZZ()
q^-1 + 79/200 + 42877/640000*q + O(q^2)
sage: MFSeriesConstructor(group=5, prec=3).J_inv_ZZ().parent()
Laurent Series Ring in q over Rational Field
sage: MFSeriesConstructor(group=infinity, prec=3).J_inv_ZZ()
q^-1 + 3/8 + 69/1024*q + O(q^2)
"""
F1 = lambda a,b: self._series_ring(
[ ZZ(0) ]
+ [
rising_factorial(a,k) * rising_factorial(b,k) / (ZZ(k).factorial())**2
* sum(ZZ(1)/(a+j) + ZZ(1)/(b+j) - ZZ(2)/ZZ(1+j)
for j in range(ZZ(0),ZZ(k))
)
for k in range(ZZ(1), ZZ(self._prec+1))
],
ZZ(self._prec+1)
)
F = lambda a,b,c: self._series_ring(
[
rising_factorial(a,k) * rising_factorial(b,k) / rising_factorial(c,k) / ZZ(k).factorial()
for k in range(ZZ(0), ZZ(self._prec+1))
],
ZZ(self._prec+1)
)
a = self._group.alpha()
b = self._group.beta()
Phi = F1(a,b) / F(a,b,ZZ(1))
q = self._series_ring.gen()
# the current implementation of power series reversion is slow
# J_inv_ZZ = ZZ(1) / ((q*Phi.exp()).reverse())
temp_f = (q*Phi.exp()).polynomial()
new_f = temp_f.revert_series(temp_f.degree()+1)
J_inv_ZZ = ZZ(1) / (new_f + O(q**(temp_f.degree()+1)))
return J_inv_ZZ
@cached_method
def f_rho_ZZ(self):
r"""
Return the rational Fourier expansion of ``f_rho``,
where the parameter ``d`` is replaced by ``1``.
.. NOTE:
The Fourier expansion of ``f_rho`` for ``d!=1``
is given by ``f_rho_ZZ(q/d)``.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.series_constructor import MFSeriesConstructor
sage: MFSeriesConstructor(prec=3).f_rho_ZZ()
1 + 5/36*q + 5/6912*q^2 + O(q^3)
sage: MFSeriesConstructor(group=5, prec=3).f_rho_ZZ()
1 + 7/100*q + 21/160000*q^2 + O(q^3)
sage: MFSeriesConstructor(group=5, prec=3).f_rho_ZZ().parent()
Power Series Ring in q over Rational Field
sage: MFSeriesConstructor(group=infinity, prec=3).f_rho_ZZ()
1
"""
q = self._series_ring.gen()
n = self.hecke_n()
if (n == infinity):
f_rho_ZZ = self._series_ring(1)
else:
temp_expr = ((-q*self.J_inv_ZZ().derivative())**2/(self.J_inv_ZZ()*(self.J_inv_ZZ()-1))).power_series()
f_rho_ZZ = (temp_expr.log()/(n-2)).exp()
return f_rho_ZZ
@cached_method
def f_i_ZZ(self):
r"""
Return the rational Fourier expansion of ``f_i``,
where the parameter ``d`` is replaced by ``1``.
.. NOTE:
The Fourier expansion of ``f_i`` for ``d!=1``
is given by ``f_i_ZZ(q/d)``.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.series_constructor import MFSeriesConstructor
sage: MFSeriesConstructor(prec=3).f_i_ZZ()
1 - 7/24*q - 77/13824*q^2 + O(q^3)
sage: MFSeriesConstructor(group=5, prec=3).f_i_ZZ()
1 - 13/40*q - 351/64000*q^2 + O(q^3)
sage: MFSeriesConstructor(group=5, prec=3).f_i_ZZ().parent()
Power Series Ring in q over Rational Field
sage: MFSeriesConstructor(group=infinity, prec=3).f_i_ZZ()
1 - 3/8*q + 3/512*q^2 + O(q^3)
"""
q = self._series_ring.gen()
n = self.hecke_n()
if (n == infinity):
f_i_ZZ = (-q*self.J_inv_ZZ().derivative()/self.J_inv_ZZ()).power_series()
else:
temp_expr = ((-q*self.J_inv_ZZ().derivative())**n/(self.J_inv_ZZ()**(n-1)*(self.J_inv_ZZ()-1))).power_series()
f_i_ZZ = (temp_expr.log()/(n-2)).exp()
return f_i_ZZ
@cached_method
def f_inf_ZZ(self):
r"""
Return the rational Fourier expansion of ``f_inf``,
where the parameter ``d`` is replaced by ``1``.
.. NOTE:
The Fourier expansion of ``f_inf`` for ``d!=1``
is given by ``d*f_inf_ZZ(q/d)``.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.series_constructor import MFSeriesConstructor
sage: MFSeriesConstructor(prec=3).f_inf_ZZ()
q - 1/72*q^2 + 7/82944*q^3 + O(q^4)
sage: MFSeriesConstructor(group=5, prec=3).f_inf_ZZ()
q - 9/200*q^2 + 279/640000*q^3 + O(q^4)
sage: MFSeriesConstructor(group=5, prec=3).f_inf_ZZ().parent()
Power Series Ring in q over Rational Field
sage: MFSeriesConstructor(group=infinity, prec=3).f_inf_ZZ()
q - 1/8*q^2 + 7/1024*q^3 + O(q^4)
"""
q = self._series_ring.gen()
n = self.hecke_n()
if (n == infinity):
f_inf_ZZ = ((-q*self.J_inv_ZZ().derivative())**2/(self.J_inv_ZZ()**2*(self.J_inv_ZZ()-1))).power_series()
else:
temp_expr = ((-q*self.J_inv_ZZ().derivative())**(2*n)/(self.J_inv_ZZ()**(2*n-2)*(self.J_inv_ZZ()-1)**n)/q**(n-2)).power_series()
f_inf_ZZ = (temp_expr.log()/(n-2)).exp()*q
return f_inf_ZZ
@cached_method
def G_inv_ZZ(self):
r"""
Return the rational Fourier expansion of ``G_inv``,
where the parameter ``d`` is replaced by ``1``.
.. NOTE:
The Fourier expansion of ``G_inv`` for ``d!=1``
is given by ``d*G_inv_ZZ(q/d)``.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.series_constructor import MFSeriesConstructor
sage: MFSeriesConstructor(group=4, prec=3).G_inv_ZZ()
q^-1 - 3/32 - 955/16384*q + O(q^2)
sage: MFSeriesConstructor(group=8, prec=3).G_inv_ZZ()
q^-1 - 15/128 - 15139/262144*q + O(q^2)
sage: MFSeriesConstructor(group=8, prec=3).G_inv_ZZ().parent()
Laurent Series Ring in q over Rational Field
sage: MFSeriesConstructor(group=infinity, prec=3).G_inv_ZZ()
q^-1 - 1/8 - 59/1024*q + O(q^2)
"""
n = self.hecke_n()
# Note that G_inv is not a weakly holomorphic form (because of the behavior at -1)
if (n == infinity):
q = self._series_ring.gen()
temp_expr = (self.J_inv_ZZ()/self.f_inf_ZZ()*q**2).power_series()
return 1/q*self.f_i_ZZ()*(temp_expr.log()/2).exp()
elif (ZZ(2).divides(n)):
return self.f_i_ZZ()*(self.f_rho_ZZ()**(ZZ(n/ZZ(2))))/self.f_inf_ZZ()
else:
#return self._qseries_ring([])
raise ValueError("G_inv doesn't exist for n={}.".format(self.hecke_n()))
@cached_method
def E4_ZZ(self):
r"""
Return the rational Fourier expansion of ``E_4``,
where the parameter ``d`` is replaced by ``1``.
.. NOTE:
The Fourier expansion of ``E4`` for ``d!=1``
is given by ``E4_ZZ(q/d)``.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.series_constructor import MFSeriesConstructor
sage: MFSeriesConstructor(prec=3).E4_ZZ()
1 + 5/36*q + 5/6912*q^2 + O(q^3)
sage: MFSeriesConstructor(group=5, prec=3).E4_ZZ()
1 + 21/100*q + 483/32000*q^2 + O(q^3)
sage: MFSeriesConstructor(group=5, prec=3).E4_ZZ().parent()
Power Series Ring in q over Rational Field
sage: MFSeriesConstructor(group=infinity, prec=3).E4_ZZ()
1 + 1/4*q + 7/256*q^2 + O(q^3)
"""
q = self._series_ring.gen()
E4_ZZ = ((-q*self.J_inv_ZZ().derivative())**2/(self.J_inv_ZZ()*(self.J_inv_ZZ()-1))).power_series()
return E4_ZZ
@cached_method
def E6_ZZ(self):
r"""
Return the rational Fourier expansion of ``E_6``,
where the parameter ``d`` is replaced by ``1``.
.. NOTE:
The Fourier expansion of ``E6`` for ``d!=1``
is given by ``E6_ZZ(q/d)``.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.series_constructor import MFSeriesConstructor
sage: MFSeriesConstructor(prec=3).E6_ZZ()
1 - 7/24*q - 77/13824*q^2 + O(q^3)
sage: MFSeriesConstructor(group=5, prec=3).E6_ZZ()
1 - 37/200*q - 14663/320000*q^2 + O(q^3)
sage: MFSeriesConstructor(group=5, prec=3).E6_ZZ().parent()
Power Series Ring in q over Rational Field
sage: MFSeriesConstructor(group=infinity, prec=3).E6_ZZ()
1 - 1/8*q - 31/512*q^2 + O(q^3)
"""
q = self._series_ring.gen()
E6_ZZ = ((-q*self.J_inv_ZZ().derivative())**3/(self.J_inv_ZZ()**2*(self.J_inv_ZZ()-1))).power_series()
return E6_ZZ
@cached_method
def Delta_ZZ(self):
r"""
Return the rational Fourier expansion of ``Delta``,
where the parameter ``d`` is replaced by ``1``.
.. NOTE:
The Fourier expansion of ``Delta`` for ``d!=1``
is given by ``d*Delta_ZZ(q/d)``.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.series_constructor import MFSeriesConstructor
sage: MFSeriesConstructor(prec=3).Delta_ZZ()
q - 1/72*q^2 + 7/82944*q^3 + O(q^4)
sage: MFSeriesConstructor(group=5, prec=3).Delta_ZZ()
q + 47/200*q^2 + 11367/640000*q^3 + O(q^4)
sage: MFSeriesConstructor(group=5, prec=3).Delta_ZZ().parent()
Power Series Ring in q over Rational Field
sage: MFSeriesConstructor(group=infinity, prec=3).Delta_ZZ()
q + 3/8*q^2 + 63/1024*q^3 + O(q^4)
"""
return (self.f_inf_ZZ()**3*self.J_inv_ZZ()**2/(self.f_rho_ZZ()**6)).power_series()
@cached_method
def E2_ZZ(self):
r"""
Return the rational Fourier expansion of ``E2``,
where the parameter ``d`` is replaced by ``1``.
.. NOTE:
The Fourier expansion of ``E2`` for ``d!=1``
is given by ``E2_ZZ(q/d)``.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.series_constructor import MFSeriesConstructor
sage: MFSeriesConstructor(prec=3).E2_ZZ()
1 - 1/72*q - 1/41472*q^2 + O(q^3)
sage: MFSeriesConstructor(group=5, prec=3).E2_ZZ()
1 - 9/200*q - 369/320000*q^2 + O(q^3)
sage: MFSeriesConstructor(group=5, prec=3).E2_ZZ().parent()
Power Series Ring in q over Rational Field
sage: MFSeriesConstructor(group=infinity, prec=3).E2_ZZ()
1 - 1/8*q - 1/512*q^2 + O(q^3)
"""
q = self._series_ring.gen()
E2_ZZ = (q*self.f_inf_ZZ().derivative())/self.f_inf_ZZ()
return E2_ZZ
@cached_method
def EisensteinSeries_ZZ(self, k):
r"""
Return the rational Fourier expansion of the normalized Eisenstein series
of weight ``k``, where the parameter ``d`` is replaced by ``1``.
Only arithmetic groups with ``n < infinity`` are supported!
.. NOTE:
THe Fourier expansion of the series is given by ``EisensteinSeries_ZZ(q/d)``.
INPUT:
- ``k`` -- A non-negative even integer, namely the weight.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.series_constructor import MFSeriesConstructor
sage: MFC = MFSeriesConstructor(prec=6)
sage: MFC.EisensteinSeries_ZZ(k=0)
1
sage: MFC.EisensteinSeries_ZZ(k=2)
1 - 1/72*q - 1/41472*q^2 - 1/53747712*q^3 - 7/371504185344*q^4 - 1/106993205379072*q^5 + O(q^6)
sage: MFC.EisensteinSeries_ZZ(k=6)
1 - 7/24*q - 77/13824*q^2 - 427/17915904*q^3 - 7399/123834728448*q^4 - 3647/35664401793024*q^5 + O(q^6)
sage: MFC.EisensteinSeries_ZZ(k=12)
1 + 455/8292*q + 310765/4776192*q^2 + 20150585/6189944832*q^3 + 1909340615/42784898678784*q^4 + 3702799555/12322050819489792*q^5 + O(q^6)
sage: MFC.EisensteinSeries_ZZ(k=12).parent()
Power Series Ring in q over Rational Field
sage: MFC = MFSeriesConstructor(group=4, prec=5)
sage: MFC.EisensteinSeries_ZZ(k=2)
1 - 1/32*q - 5/8192*q^2 - 1/524288*q^3 - 13/536870912*q^4 + O(q^5)
sage: MFC.EisensteinSeries_ZZ(k=4)
1 + 3/16*q + 39/4096*q^2 + 21/262144*q^3 + 327/268435456*q^4 + O(q^5)
sage: MFC.EisensteinSeries_ZZ(k=6)
1 - 7/32*q - 287/8192*q^2 - 427/524288*q^3 - 9247/536870912*q^4 + O(q^5)
sage: MFC.EisensteinSeries_ZZ(k=12)
1 + 63/11056*q + 133119/2830336*q^2 + 2790081/181141504*q^3 + 272631807/185488900096*q^4 + O(q^5)
sage: MFC = MFSeriesConstructor(group=6, prec=5)
sage: MFC.EisensteinSeries_ZZ(k=2)
1 - 1/18*q - 1/648*q^2 - 7/209952*q^3 - 7/22674816*q^4 + O(q^5)
sage: MFC.EisensteinSeries_ZZ(k=4)
1 + 2/9*q + 1/54*q^2 + 37/52488*q^3 + 73/5668704*q^4 + O(q^5)
sage: MFC.EisensteinSeries_ZZ(k=6)
1 - 1/6*q - 11/216*q^2 - 271/69984*q^3 - 1057/7558272*q^4 + O(q^5)
sage: MFC.EisensteinSeries_ZZ(k=12)
1 + 182/151329*q + 62153/2723922*q^2 + 16186807/882550728*q^3 + 381868123/95315478624*q^4 + O(q^5)
"""
try:
if k < 0:
raise TypeError(None)
k = 2*ZZ(k/2)
except TypeError:
raise TypeError("k={} has to be a non-negative even integer!".format(k))
if (not self.group().is_arithmetic() or self.group().n() == infinity):
# Exceptional cases should be called manually (see in FormsRing_abstract)
raise NotImplementedError("Eisenstein series are only supported in the finite arithmetic cases!")
# Trivial case
if k == 0:
return self._series_ring(1)
M = ZZ(self.group().lam()**2)
lamk = M**(ZZ(k/2))
dval = self.group().dvalue()
def coeff(m):
m = ZZ(m)
if m < 0:
return ZZ(0)
elif m == 0:
return ZZ(1)
factor = -2*k / QQ(bernoulli(k)) / lamk
sum1 = sigma(m, k-1)
if M.divides(m):
sum2 = (lamk-1) * sigma(ZZ(m/M), k-1)
else:
sum2 = ZZ(0)
if (M == 1):
sum3 = ZZ(0)
else:
if (m == 1):
N = ZZ(1)
else:
N = ZZ(m / M**ZZ(m.valuation(M)))
sum3 = -sigma(ZZ(N), k-1) * ZZ(m/N)**(k-1) / (lamk + 1)
return factor * (sum1 + sum2 + sum3) * dval**m
q = self._series_ring.gen()
return sum([coeff(m)*q**m for m in range(self.prec())]).add_bigoh(self.prec())
class MFSeriesConstructor(SageObject, UniqueRepresentation):
r"""
Constructor for the Fourier expansion of some
(specific, basic) modular forms.
The constructor is used by forms elements in case
their Fourier expansion is needed or requested.
"""
@staticmethod
def __classcall__(cls, group=HeckeTriangleGroup(3), prec=ZZ(10)):
r"""
Return a (cached) instance with canonical parameters.
.. NOTE:
For each choice of group and precision the constructor is
cached (only) once. Further calculations with different
base rings and possibly numerical parameters are based on
the same cached instance.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.series_constructor import MFSeriesConstructor
sage: MFSeriesConstructor() == MFSeriesConstructor(3, 10)
True
sage: MFSeriesConstructor(group=4).hecke_n()
4
sage: MFSeriesConstructor(group=5, prec=12).prec()
12
"""
if (group == infinity):
group = HeckeTriangleGroup(infinity)
else:
try:
group = HeckeTriangleGroup(ZZ(group))
except TypeError:
group = HeckeTriangleGroup(group.n())
prec = ZZ(prec)
# We don't need this assumption the precision may in principle also be negative.
# if (prec<1):
# raise Exception("prec must be an Integer >=1")
return super(MFSeriesConstructor, cls).__classcall__(cls, group, prec)
def __init__(self, group, prec):
r"""
Constructor for the Fourier expansion of some
(specific, basic) modular forms.
INPUT:
- ``group`` -- A Hecke triangle group (default: HeckeTriangleGroup(3)).
- ``prec`` -- An integer (default: 10), the default precision used
in calculations in the LaurentSeriesRing or PowerSeriesRing.
OUTPUT:
The constructor for Fourier expansion with the specified settings.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.series_constructor import MFSeriesConstructor
sage: MFC = MFSeriesConstructor()
sage: MFC
Power series constructor for Hecke modular forms for n=3 with (basic series) precision 10
sage: MFC.group()
Hecke triangle group for n = 3
sage: MFC.prec()
10
sage: MFC._series_ring
Power Series Ring in q over Rational Field
sage: MFSeriesConstructor(group=4)
Power series constructor for Hecke modular forms for n=4 with (basic series) precision 10
sage: MFSeriesConstructor(group=5, prec=12)
Power series constructor for Hecke modular forms for n=5 with (basic series) precision 12
sage: MFSeriesConstructor(group=infinity)
Power series constructor for Hecke modular forms for n=+Infinity with (basic series) precision 10
"""
self._group = group
self._prec = prec
self._series_ring = PowerSeriesRing(QQ, 'q', default_prec=self._prec)
def _repr_(self):
r"""
Return the string representation of ``self``.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.series_constructor import MFSeriesConstructor
sage: MFSeriesConstructor(group=4)
Power series constructor for Hecke modular forms for n=4 with (basic series) precision 10
sage: MFSeriesConstructor(group=5, prec=12)
Power series constructor for Hecke modular forms for n=5 with (basic series) precision 12
"""
return "Power series constructor for Hecke modular forms for n={} with (basic series) precision {}".\
format(self._group.n(), self._prec)
def group(self):
r"""
Return the (Hecke triangle) group of ``self``.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.series_constructor import MFSeriesConstructor
sage: MFSeriesConstructor(group=4).group()
Hecke triangle group for n = 4
"""
return self._group
def hecke_n(self):
r"""
Return the parameter ``n`` of the (Hecke triangle) group of ``self``.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.series_constructor import MFSeriesConstructor
sage: MFSeriesConstructor(group=4).hecke_n()
4
"""
return self._group.n()
def prec(self):
r"""
Return the used default precision for the PowerSeriesRing or LaurentSeriesRing.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.series_constructor import MFSeriesConstructor
sage: MFSeriesConstructor(group=5).prec()
10
sage: MFSeriesConstructor(group=5, prec=20).prec()
20
"""
return self._prec
@cached_method
def J_inv_ZZ(self):
r"""
Return the rational Fourier expansion of ``J_inv``,
where the parameter ``d`` is replaced by ``1``.
This is the main function used to determine all Fourier expansions!
.. NOTE:
The Fourier expansion of ``J_inv`` for ``d!=1``
is given by ``J_inv_ZZ(q/d)``.
.. TODO:
The functions that are used in this implementation are
products of hypergeometric series with other, elementary,
functions. Implement them and clean up this representation.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.series_constructor import MFSeriesConstructor
sage: MFSeriesConstructor(prec=3).J_inv_ZZ()
q^-1 + 31/72 + 1823/27648*q + O(q^2)
sage: MFSeriesConstructor(group=5, prec=3).J_inv_ZZ()
q^-1 + 79/200 + 42877/640000*q + O(q^2)
sage: MFSeriesConstructor(group=5, prec=3).J_inv_ZZ().parent()
Laurent Series Ring in q over Rational Field
sage: MFSeriesConstructor(group=infinity, prec=3).J_inv_ZZ()
q^-1 + 3/8 + 69/1024*q + O(q^2)
"""
F1 = lambda a, b: self._series_ring([ZZ(0)] + [
rising_factorial(a, k) * rising_factorial(b, k) /
(ZZ(k).factorial())**2 * sum(
ZZ(1) / (a + j) + ZZ(1) / (b + j) - ZZ(2) / ZZ(1 + j)
for j in range(ZZ(0), ZZ(k)))
for k in range(ZZ(1), ZZ(self._prec + 1))
], ZZ(self._prec + 1))
F = lambda a, b, c: self._series_ring([
rising_factorial(a, k) * rising_factorial(b, k) / rising_factorial(
c, k) / ZZ(k).factorial()
for k in range(ZZ(0), ZZ(self._prec + 1))
], ZZ(self._prec + 1))
a = self._group.alpha()
b = self._group.beta()
Phi = F1(a, b) / F(a, b, ZZ(1))
q = self._series_ring.gen()
# the current implementation of power series reversion is slow
# J_inv_ZZ = ZZ(1) / ((q*Phi.exp()).reverse())
temp_f = (q * Phi.exp()).polynomial()
new_f = temp_f.revert_series(temp_f.degree() + 1)
J_inv_ZZ = ZZ(1) / (new_f + O(q**(temp_f.degree() + 1)))
return J_inv_ZZ
@cached_method
def f_rho_ZZ(self):
r"""
Return the rational Fourier expansion of ``f_rho``,
where the parameter ``d`` is replaced by ``1``.
.. NOTE:
The Fourier expansion of ``f_rho`` for ``d!=1``
is given by ``f_rho_ZZ(q/d)``.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.series_constructor import MFSeriesConstructor
sage: MFSeriesConstructor(prec=3).f_rho_ZZ()
1 + 5/36*q + 5/6912*q^2 + O(q^3)
sage: MFSeriesConstructor(group=5, prec=3).f_rho_ZZ()
1 + 7/100*q + 21/160000*q^2 + O(q^3)
sage: MFSeriesConstructor(group=5, prec=3).f_rho_ZZ().parent()
Power Series Ring in q over Rational Field
sage: MFSeriesConstructor(group=infinity, prec=3).f_rho_ZZ()
1
"""
q = self._series_ring.gen()
n = self.hecke_n()
if (n == infinity):
f_rho_ZZ = self._series_ring(1)
else:
temp_expr = ((-q * self.J_inv_ZZ().derivative())**2 /
(self.J_inv_ZZ() *
(self.J_inv_ZZ() - 1))).power_series()
f_rho_ZZ = (temp_expr.log() / (n - 2)).exp()
return f_rho_ZZ
@cached_method
def f_i_ZZ(self):
r"""
Return the rational Fourier expansion of ``f_i``,
where the parameter ``d`` is replaced by ``1``.
.. NOTE:
The Fourier expansion of ``f_i`` for ``d!=1``
is given by ``f_i_ZZ(q/d)``.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.series_constructor import MFSeriesConstructor
sage: MFSeriesConstructor(prec=3).f_i_ZZ()
1 - 7/24*q - 77/13824*q^2 + O(q^3)
sage: MFSeriesConstructor(group=5, prec=3).f_i_ZZ()
1 - 13/40*q - 351/64000*q^2 + O(q^3)
sage: MFSeriesConstructor(group=5, prec=3).f_i_ZZ().parent()
Power Series Ring in q over Rational Field
sage: MFSeriesConstructor(group=infinity, prec=3).f_i_ZZ()
1 - 3/8*q + 3/512*q^2 + O(q^3)
"""
q = self._series_ring.gen()
n = self.hecke_n()
if (n == infinity):
f_i_ZZ = (-q * self.J_inv_ZZ().derivative() /
self.J_inv_ZZ()).power_series()
else:
temp_expr = ((-q * self.J_inv_ZZ().derivative())**n /
(self.J_inv_ZZ()**(n - 1) *
(self.J_inv_ZZ() - 1))).power_series()
f_i_ZZ = (temp_expr.log() / (n - 2)).exp()
return f_i_ZZ
@cached_method
def f_inf_ZZ(self):
r"""
Return the rational Fourier expansion of ``f_inf``,
where the parameter ``d`` is replaced by ``1``.
.. NOTE:
The Fourier expansion of ``f_inf`` for ``d!=1``
is given by ``d*f_inf_ZZ(q/d)``.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.series_constructor import MFSeriesConstructor
sage: MFSeriesConstructor(prec=3).f_inf_ZZ()
q - 1/72*q^2 + 7/82944*q^3 + O(q^4)
sage: MFSeriesConstructor(group=5, prec=3).f_inf_ZZ()
q - 9/200*q^2 + 279/640000*q^3 + O(q^4)
sage: MFSeriesConstructor(group=5, prec=3).f_inf_ZZ().parent()
Power Series Ring in q over Rational Field
sage: MFSeriesConstructor(group=infinity, prec=3).f_inf_ZZ()
q - 1/8*q^2 + 7/1024*q^3 + O(q^4)
"""
q = self._series_ring.gen()
n = self.hecke_n()
if (n == infinity):
f_inf_ZZ = ((-q * self.J_inv_ZZ().derivative())**2 /
(self.J_inv_ZZ()**2 *
(self.J_inv_ZZ() - 1))).power_series()
else:
temp_expr = ((-q * self.J_inv_ZZ().derivative())**(2 * n) /
(self.J_inv_ZZ()**(2 * n - 2) *
(self.J_inv_ZZ() - 1)**n) /
q**(n - 2)).power_series()
f_inf_ZZ = (temp_expr.log() / (n - 2)).exp() * q
return f_inf_ZZ
@cached_method
def G_inv_ZZ(self):
r"""
Return the rational Fourier expansion of ``G_inv``,
where the parameter ``d`` is replaced by ``1``.
.. NOTE:
The Fourier expansion of ``G_inv`` for ``d!=1``
is given by ``d*G_inv_ZZ(q/d)``.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.series_constructor import MFSeriesConstructor
sage: MFSeriesConstructor(group=4, prec=3).G_inv_ZZ()
q^-1 - 3/32 - 955/16384*q + O(q^2)
sage: MFSeriesConstructor(group=8, prec=3).G_inv_ZZ()
q^-1 - 15/128 - 15139/262144*q + O(q^2)
sage: MFSeriesConstructor(group=8, prec=3).G_inv_ZZ().parent()
Laurent Series Ring in q over Rational Field
sage: MFSeriesConstructor(group=infinity, prec=3).G_inv_ZZ()
q^-1 - 1/8 - 59/1024*q + O(q^2)
"""
n = self.hecke_n()
# Note that G_inv is not a weakly holomorphic form (because of the behavior at -1)
if (n == infinity):
q = self._series_ring.gen()
temp_expr = (self.J_inv_ZZ() / self.f_inf_ZZ() *
q**2).power_series()
return 1 / q * self.f_i_ZZ() * (temp_expr.log() / 2).exp()
elif (ZZ(2).divides(n)):
return self.f_i_ZZ() * (self.f_rho_ZZ()**(ZZ(
n / ZZ(2)))) / self.f_inf_ZZ()
else:
#return self._qseries_ring([])
raise ValueError("G_inv doesn't exist for n={}.".format(
self.hecke_n()))
@cached_method
def E4_ZZ(self):
r"""
Return the rational Fourier expansion of ``E_4``,
where the parameter ``d`` is replaced by ``1``.
.. NOTE:
The Fourier expansion of ``E4`` for ``d!=1``
is given by ``E4_ZZ(q/d)``.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.series_constructor import MFSeriesConstructor
sage: MFSeriesConstructor(prec=3).E4_ZZ()
1 + 5/36*q + 5/6912*q^2 + O(q^3)
sage: MFSeriesConstructor(group=5, prec=3).E4_ZZ()
1 + 21/100*q + 483/32000*q^2 + O(q^3)
sage: MFSeriesConstructor(group=5, prec=3).E4_ZZ().parent()
Power Series Ring in q over Rational Field
sage: MFSeriesConstructor(group=infinity, prec=3).E4_ZZ()
1 + 1/4*q + 7/256*q^2 + O(q^3)
"""
q = self._series_ring.gen()
E4_ZZ = ((-q * self.J_inv_ZZ().derivative())**2 /
(self.J_inv_ZZ() * (self.J_inv_ZZ() - 1))).power_series()
return E4_ZZ
@cached_method
def E6_ZZ(self):
r"""
Return the rational Fourier expansion of ``E_6``,
where the parameter ``d`` is replaced by ``1``.
.. NOTE:
The Fourier expansion of ``E6`` for ``d!=1``
is given by ``E6_ZZ(q/d)``.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.series_constructor import MFSeriesConstructor
sage: MFSeriesConstructor(prec=3).E6_ZZ()
1 - 7/24*q - 77/13824*q^2 + O(q^3)
sage: MFSeriesConstructor(group=5, prec=3).E6_ZZ()
1 - 37/200*q - 14663/320000*q^2 + O(q^3)
sage: MFSeriesConstructor(group=5, prec=3).E6_ZZ().parent()
Power Series Ring in q over Rational Field
sage: MFSeriesConstructor(group=infinity, prec=3).E6_ZZ()
1 - 1/8*q - 31/512*q^2 + O(q^3)
"""
q = self._series_ring.gen()
E6_ZZ = ((-q * self.J_inv_ZZ().derivative())**3 /
(self.J_inv_ZZ()**2 * (self.J_inv_ZZ() - 1))).power_series()
return E6_ZZ
@cached_method
def Delta_ZZ(self):
r"""
Return the rational Fourier expansion of ``Delta``,
where the parameter ``d`` is replaced by ``1``.
.. NOTE:
The Fourier expansion of ``Delta`` for ``d!=1``
is given by ``d*Delta_ZZ(q/d)``.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.series_constructor import MFSeriesConstructor
sage: MFSeriesConstructor(prec=3).Delta_ZZ()
q - 1/72*q^2 + 7/82944*q^3 + O(q^4)
sage: MFSeriesConstructor(group=5, prec=3).Delta_ZZ()
q + 47/200*q^2 + 11367/640000*q^3 + O(q^4)
sage: MFSeriesConstructor(group=5, prec=3).Delta_ZZ().parent()
Power Series Ring in q over Rational Field
sage: MFSeriesConstructor(group=infinity, prec=3).Delta_ZZ()
q + 3/8*q^2 + 63/1024*q^3 + O(q^4)
"""
return (self.f_inf_ZZ()**3 * self.J_inv_ZZ()**2 /
(self.f_rho_ZZ()**6)).power_series()
@cached_method
def E2_ZZ(self):
r"""
Return the rational Fourier expansion of ``E2``,
where the parameter ``d`` is replaced by ``1``.
.. NOTE:
The Fourier expansion of ``E2`` for ``d!=1``
is given by ``E2_ZZ(q/d)``.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.series_constructor import MFSeriesConstructor
sage: MFSeriesConstructor(prec=3).E2_ZZ()
1 - 1/72*q - 1/41472*q^2 + O(q^3)
sage: MFSeriesConstructor(group=5, prec=3).E2_ZZ()
1 - 9/200*q - 369/320000*q^2 + O(q^3)
sage: MFSeriesConstructor(group=5, prec=3).E2_ZZ().parent()
Power Series Ring in q over Rational Field
sage: MFSeriesConstructor(group=infinity, prec=3).E2_ZZ()
1 - 1/8*q - 1/512*q^2 + O(q^3)
"""
q = self._series_ring.gen()
E2_ZZ = (q * self.f_inf_ZZ().derivative()) / self.f_inf_ZZ()
return E2_ZZ
@cached_method
def EisensteinSeries_ZZ(self, k):
r"""
Return the rational Fourier expansion of the normalized Eisenstein series
of weight ``k``, where the parameter ``d`` is replaced by ``1``.
Only arithmetic groups with ``n < infinity`` are supported!
.. NOTE:
THe Fourier expansion of the series is given by ``EisensteinSeries_ZZ(q/d)``.
INPUT:
- ``k`` -- A non-negative even integer, namely the weight.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.series_constructor import MFSeriesConstructor
sage: MFC = MFSeriesConstructor(prec=6)
sage: MFC.EisensteinSeries_ZZ(k=0)
1
sage: MFC.EisensteinSeries_ZZ(k=2)
1 - 1/72*q - 1/41472*q^2 - 1/53747712*q^3 - 7/371504185344*q^4 - 1/106993205379072*q^5 + O(q^6)
sage: MFC.EisensteinSeries_ZZ(k=6)
1 - 7/24*q - 77/13824*q^2 - 427/17915904*q^3 - 7399/123834728448*q^4 - 3647/35664401793024*q^5 + O(q^6)
sage: MFC.EisensteinSeries_ZZ(k=12)
1 + 455/8292*q + 310765/4776192*q^2 + 20150585/6189944832*q^3 + 1909340615/42784898678784*q^4 + 3702799555/12322050819489792*q^5 + O(q^6)
sage: MFC.EisensteinSeries_ZZ(k=12).parent()
Power Series Ring in q over Rational Field
sage: MFC = MFSeriesConstructor(group=4, prec=5)
sage: MFC.EisensteinSeries_ZZ(k=2)
1 - 1/32*q - 5/8192*q^2 - 1/524288*q^3 - 13/536870912*q^4 + O(q^5)
sage: MFC.EisensteinSeries_ZZ(k=4)
1 + 3/16*q + 39/4096*q^2 + 21/262144*q^3 + 327/268435456*q^4 + O(q^5)
sage: MFC.EisensteinSeries_ZZ(k=6)
1 - 7/32*q - 287/8192*q^2 - 427/524288*q^3 - 9247/536870912*q^4 + O(q^5)
sage: MFC.EisensteinSeries_ZZ(k=12)
1 + 63/11056*q + 133119/2830336*q^2 + 2790081/181141504*q^3 + 272631807/185488900096*q^4 + O(q^5)
sage: MFC = MFSeriesConstructor(group=6, prec=5)
sage: MFC.EisensteinSeries_ZZ(k=2)
1 - 1/18*q - 1/648*q^2 - 7/209952*q^3 - 7/22674816*q^4 + O(q^5)
sage: MFC.EisensteinSeries_ZZ(k=4)
1 + 2/9*q + 1/54*q^2 + 37/52488*q^3 + 73/5668704*q^4 + O(q^5)
sage: MFC.EisensteinSeries_ZZ(k=6)
1 - 1/6*q - 11/216*q^2 - 271/69984*q^3 - 1057/7558272*q^4 + O(q^5)
sage: MFC.EisensteinSeries_ZZ(k=12)
1 + 182/151329*q + 62153/2723922*q^2 + 16186807/882550728*q^3 + 381868123/95315478624*q^4 + O(q^5)
"""
try:
if k < 0:
raise TypeError(None)
k = 2 * ZZ(k / 2)
except TypeError:
raise TypeError(
"k={} has to be a non-negative even integer!".format(k))
if (not self.group().is_arithmetic() or self.group().n() == infinity):
# Exceptional cases should be called manually (see in FormsRing_abstract)
raise NotImplementedError(
"Eisenstein series are only supported in the finite arithmetic cases!"
)
# Trivial case
if k == 0:
return self._series_ring(1)
M = ZZ(self.group().lam()**2)
lamk = M**(ZZ(k / 2))
dval = self.group().dvalue()
def coeff(m):
m = ZZ(m)
if m < 0:
return ZZ(0)
elif m == 0:
return ZZ(1)
factor = -2 * k / QQ(bernoulli(k)) / lamk
sum1 = sigma(m, k - 1)
if M.divides(m):
sum2 = (lamk - 1) * sigma(ZZ(m / M), k - 1)
else:
sum2 = ZZ(0)
if (M == 1):
sum3 = ZZ(0)
else:
if (m == 1):
N = ZZ(1)
else:
N = ZZ(m / M**ZZ(m.valuation(M)))
sum3 = -sigma(ZZ(N), k - 1) * ZZ(m / N)**(k - 1) / (lamk + 1)
return factor * (sum1 + sum2 + sum3) * dval**m
q = self._series_ring.gen()
return sum([coeff(m) * q**m
for m in range(self.prec())]).add_bigoh(self.prec())
def series(self, n=2, quadratic_twist=+1, prec=5):
r"""
Returns the `n`-th approximation to the `p`-adic L-series as a
power series in `T` (corresponding to `\gamma-1` with
`\gamma=1+p` as a generator of `1+p\ZZ_p`). Each coefficient
is a `p`-adic number whose precision is provably correct.
Here the normalization of the `p`-adic L-series is chosen such
that `L_p(J,1) = (1-1/\alpha)^2 L(J,1)/\Omega_J` where
`\alpha` is the unit root
INPUT:
- ``n`` - (default: 2) a positive integer
- ``quadratic_twist`` - (default: +1) a fundamental
discriminant of a quadratic field, coprime to the
conductor of the curve
- ``prec`` - (default: 5) maximal number of terms of the
series to compute; to compute as many as possible just
give a very large number for ``prec``; the result will
still be correct.
ALIAS: power_series is identical to series.
EXAMPLES:
sage: J = J0(188)[0]
sage: p = 7
sage: L = J.padic_lseries(p)
sage: L.is_ordinary()
True
sage: f = L.series(2)
sage: f[0]
O(7^20)
sage: f[1].norm()
3 + 4*7 + 3*7^2 + 6*7^3 + 5*7^4 + 5*7^5 + 6*7^6 + 4*7^7 + 5*7^8 + 7^10 + 5*7^11 + 4*7^13 + 4*7^14 + 5*7^15 + 2*7^16 + 5*7^17 + 7^18 + 7^19 + O(7^20)
"""
n = ZZ(n)
if n < 1:
raise ValueError, "n (=%s) must be a positive integer"%n
if not self.is_ordinary():
raise ValueError, "p (=%s) must be an ordinary prime"%p
# check if the conditions on quadratic_twist are satisfied
D = ZZ(quadratic_twist)
if D != 1:
if D % 4 == 0:
d = D//4
if not d.is_squarefree() or d % 4 == 1:
raise ValueError, "quadratic_twist (=%s) must be a fundamental discriminant of a quadratic field"%D
else:
if not D.is_squarefree() or D % 4 != 1:
raise ValueError, "quadratic_twist (=%s) must be a fundamental discriminant of a quadratic field"%D
if gcd(D,self._p) != 1:
raise ValueError, "quadratic twist (=%s) must be coprime to p (=%s) "%(D,self._p)
if gcd(D,self._E.conductor())!= 1:
for ell in prime_divisors(D):
if valuation(self._E.conductor(),ell) > valuation(D,ell) :
raise ValueError, "can not twist a curve of conductor (=%s) by the quadratic twist (=%s)."%(self._E.conductor(),D)
p = self._p
if p == 2 and self._normalize :
print 'Warning : For p=2 the normalization might not be correct !'
#verbose("computing L-series for p=%s, n=%s, and prec=%s"%(p,n,prec))
# bounds = self._prec_bounds(n,prec)
# padic_prec = max(bounds[1:]) + 5
padic_prec = 10
# verbose("using p-adic precision of %s"%padic_prec)
res_series_prec = min(p**(n-1), prec)
verbose("using series precision of %s"%res_series_prec)
ans = self._get_series_from_cache(n, res_series_prec,D)
if not ans is None:
verbose("found series in cache")
return ans
K = QQ
gamma = K(1 + p)
R = PowerSeriesRing(K,'T',res_series_prec)
T = R(R.gen(),res_series_prec )
#L = R(0)
one_plus_T_factor = R(1)
gamma_power = K(1)
teich = self.teichmuller(padic_prec)
p_power = p**(n-1)
# F = Qp(p,padic_prec)
verbose("Now iterating over %s summands"%((p-1)*p_power))
verbose_level = get_verbose()
count_verb = 0
alphas = self.alpha()
#print len(alphas)
Lprod = []
self._emb = 0
if len(alphas) == 2:
split = True
else:
split = False
for alpha in alphas:
L = R(0)
self._emb = self._emb + 1
for j in range(p_power):
s = K(0)
if verbose_level >= 2 and j/p_power*100 > count_verb + 3:
verbose("%.2f percent done"%(float(j)/p_power*100))
count_verb += 3
for a in range(1,p):
if split:
# b = ((F.teichmuller(a)).lift() % ZZ(p**n))
b = (teich[a]) % ZZ(p**n)
b = b*gamma_power
else:
b = teich[a] * gamma_power
s += self.measure(b, n, padic_prec,D,alpha)
L += s * one_plus_T_factor
one_plus_T_factor *= 1+T
gamma_power *= gamma
Lprod = Lprod + [L]
if len(Lprod)==1:
return Lprod[0]
else:
return Lprod[0]*Lprod[1]
class MFSeriesConstructor(SageObject,UniqueRepresentation):
r"""
Constructor for the Fourier expansion of some
(specific, basic) modular forms.
The constructor is used by forms elements in case
their Fourier expansion is needed or requested.
"""
@staticmethod
def __classcall__(cls, group = HeckeTriangleGroup(3), base_ring = ZZ, prec=ZZ(10), fix_d=False, set_d=None, d_num_prec=ZZ(53)):
r"""
Return a (cached) instance with canonical parameters.
In particular in case ``fix_d = True`` or if ``set_d`` is
set then the ``base_ring`` is replaced by the common parent
of ``base_ring`` and the parent of ``set_d`` (resp. the
numerical value of ``d`` in case ``fix_d=True``).
EXAMPLES::
sage: MFSeriesConstructor() == MFSeriesConstructor(3, ZZ, 10, False, None, 53)
True
sage: MFSeriesConstructor(base_ring = CC, set_d=CC(1)) == MFSeriesConstructor(set_d=CC(1))
True
sage: MFSeriesConstructor(group=4, fix_d=True).base_ring() == QQ
True
sage: MFSeriesConstructor(group=5, fix_d=True).base_ring() == RR
True
"""
if (group==infinity):
group = HeckeTriangleGroup(infinity)
else:
try:
group = HeckeTriangleGroup(ZZ(group))
except TypeError:
group = HeckeTriangleGroup(group.n())
prec=ZZ(prec)
#if (prec<1):
# raise Exception("prec must be an Integer >=1")
fix_d = bool(fix_d)
if (fix_d):
n = group.n()
d = group.dvalue()
if (group.is_arithmetic()):
d_num_prec = None
set_d = 1/base_ring(1/d)
else:
d_num_prec = ZZ(d_num_prec)
set_d = group.dvalue().n(d_num_prec)
else:
d_num_prec = None
if (set_d is not None):
base_ring=(base_ring(1)*set_d).parent()
#elif (not base_ring.is_exact()):
# raise NotImplementedError
return super(MFSeriesConstructor,cls).__classcall__(cls, group, base_ring, prec, fix_d, set_d, d_num_prec)
def __init__(self, group, base_ring, prec, fix_d, set_d, d_num_prec):
r"""
Constructor for the Fourier expansion of some
(specific, basic) modular forms.
INPUT:
- ``group`` - A Hecke triangle group (default: HeckeTriangleGroup(3)).
- ``base_ring`` - The base ring (default: ZZ)
- ``prec`` - An integer (default: 10), the default precision used
in calculations in the LaurentSeriesRing or PowerSeriesRing.
- ``fix_d`` - ``True`` or ``False`` (default: ``False``).
If ``fix_d == False`` the base ring of the power series
is (the fraction field) of the polynomial ring over the base
ring in one formal parameter ``d``.
If ``fix_d == True`` the formal parameter ``d`` is replaced
by its numerical value with numerical precision at least ``d_num_prec``
(or exact in case n=3, 4, 6). The base ring of the PowerSeriesRing
or LaurentSeriesRing is changed to a common parent of
``base_ring`` and the parent of the mentioned value ``d``.
- ``set_d`` - A number which replaces the formal parameter ``d``.
The base ring of the PowerSeriesRing or LaurentSeriesRing is
changed to a common parent of ``base_ring``
and the parent of the specified value for ``d``.
Note that in particular ``set_d=1`` will produce
rational Fourier expansions.
- ``d_num_prec`` - An integer, a lower bound for the precision of the
numerical value of ``d``.
OUTPUT:
The constructor for Fourier expansion with the specified settings.
EXAMPLES::
sage: MFC = MFSeriesConstructor()
sage: MFC
Power series constructor for Hecke modular forms for n=3, base ring=Integer Ring
with (basic series) precision 10 with formal parameter d
sage: MFC.group()
Hecke triangle group for n = 3
sage: MFC.prec()
10
sage: MFC.d().parent()
Fraction Field of Univariate Polynomial Ring in d over Integer Ring
sage: MFC._ZZseries_ring
Power Series Ring in q over Rational Field
sage: MFSeriesConstructor(set_d=CC(1))
Power series constructor for Hecke modular forms for n=3, base ring=Complex Field with 53 bits of precision
with (basic series) precision 10 with parameter d=1.00000000000000
sage: MFSeriesConstructor(group=4, fix_d=True)
Power series constructor for Hecke modular forms for n=4, base ring=Rational Field
with (basic series) precision 10 with parameter d=1/256
sage: MFSeriesConstructor(group=5, fix_d=True)
Power series constructor for Hecke modular forms for n=5, base ring=Real Field with 53 bits of precision
with (basic series) precision 10 with parameter d=0.00705223418128563
"""
self._group = group
self._base_ring = base_ring
self._prec = prec
self._fix_d = fix_d
self._set_d = set_d
self._d_num_prec = d_num_prec
if (set_d):
self._coeff_ring = FractionField(base_ring)
self._d = set_d
else:
self._coeff_ring = FractionField(PolynomialRing(base_ring,"d"))
self._d = self._coeff_ring.gen()
self._ZZseries_ring = PowerSeriesRing(QQ,'q',default_prec=self._prec)
self._qseries_ring = PowerSeriesRing(self._coeff_ring,'q',default_prec=self._prec)
def _repr_(self):
r"""
Return the string representation of ``self``.
EXAMPLES::
sage: MFSeriesConstructor(group=4, fix_d=True)
Power series constructor for Hecke modular forms for n=4, base ring=Rational Field
with (basic series) precision 10 with parameter d=1/256
sage: MFSeriesConstructor(group=5)
Power series constructor for Hecke modular forms for n=5, base ring=Integer Ring
with (basic series) precision 10 with formal parameter d
"""
if (self._set_d):
return "Power series constructor for Hecke modular forms for n={}, base ring={} with (basic series) precision {} with parameter d={}".\
format(self._group.n(), self._base_ring, self._prec, self._d)
else:
return "Power series constructor for Hecke modular forms for n={}, base ring={} with (basic series) precision {} with formal parameter d".\
format(self._group.n(), self._base_ring, self._prec)
def group(self):
r"""
Return the (Hecke triangle) group of ``self``.
EXAMPLES::
sage: MFSeriesConstructor(group=4, fix_d=True).group()
Hecke triangle group for n = 4
"""
return self._group
def hecke_n(self):
r"""
Return the parameter ``n`` of the (Hecke triangle) group of ``self``.
EXAMPLES::
sage: MFSeriesConstructor(group=4, fix_d=True).hecke_n()
4
"""
return self._group.n()
def base_ring(self):
r"""
Return base ring of ``self``.
EXAMPLES::
sage: MFSeriesConstructor(group=5, fix_d=True).base_ring()
Real Field with 53 bits of precision
sage: MFSeriesConstructor(group=5, fix_d=True, d_num_prec=100).base_ring()
Real Field with 100 bits of precision
"""
return self._base_ring
def prec(self):
r"""
Return the used default precision for the PowerSeriesRing or LaurentSeriesRing.
EXAMPLES::
sage: MFSeriesConstructor(group=5, fix_d=True).prec()
10
sage: MFSeriesConstructor(group=5, prec=20).prec()
20
"""
return self._prec
def fix_d(self):
r"""
Return whether the numerical value for the parameter
``d`` will be substituted or not.
Note: Depending on whether ``set_d`` is ``None`` or
not ``d`` might still be substituted despite ``fix_d``
being ``False``.
EXAMPLES::
sage: MFSeriesConstructor(group=5, fix_d=True, set_d=1).fix_d()
True
sage: MFSeriesConstructor(group=5, fix_d=True, set_d=1).set_d()
0.00705223418128563
sage: MFSeriesConstructor(group=5, set_d=1).fix_d()
False
"""
return self._fix_d
def set_d(self):
r"""
Return the numerical value which is substituted for
the parameter ``d``. Default: ``None``, meaning
the formal parameter ``d`` is used.
EXAMPLES::
sage: MFSeriesConstructor(group=5, fix_d=True, set_d=1).set_d()
0.00705223418128563
sage: MFSeriesConstructor(group=5, set_d=1).set_d()
1
sage: MFSeriesConstructor(group=5, set_d=1).set_d().parent()
Integer Ring
"""
return self._set_d
def is_exact(self):
r"""
Return whether used ``base_ring`` is exact.
EXAMPLES::
sage: MFSeriesConstructor(group=4, fix_d=True).is_exact()
True
sage: MFSeriesConstructor(group=5, fix_d=True).is_exact()
False
sage: MFSeriesConstructor(group=5, set_d=1).is_exact()
True
"""
return self._base_ring.is_exact()
def d(self):
r"""
Return the formal parameter ``d`` respectively
its (possibly numerical) value in case ``set_d``
is not ``None``.
EXAMPLES::
sage: MFSeriesConstructor(group=4, fix_d=True).d()
1/256
sage: MFSeriesConstructor(group=4).d()
d
sage: MFSeriesConstructor(group=4).d().parent()
Fraction Field of Univariate Polynomial Ring in d over Integer Ring
sage: MFSeriesConstructor(group=5, fix_d=True).d()
0.00705223418128563
sage: MFSeriesConstructor(group=5, set_d=1).d()
1
"""
return self._d
def q(self):
r"""
Return the generator of the used PowerSeriesRing.
EXAMPLES::
sage: MFSeriesConstructor(group=4, fix_d=True).q()
q
sage: MFSeriesConstructor(group=4, fix_d=True).q().parent()
Power Series Ring in q over Rational Field
sage: MFSeriesConstructor(group=5, fix_d=True).q().parent()
Power Series Ring in q over Real Field with 53 bits of precision
"""
return self._qseries_ring.gen()
def coeff_ring(self):
r"""
Return coefficient ring of ``self``.
EXAMPLES::
sage: MFSeriesConstructor(group=4, fix_d=True).coeff_ring()
Rational Field
sage: MFSeriesConstructor(group=4).coeff_ring()
Fraction Field of Univariate Polynomial Ring in d over Integer Ring
sage: MFSeriesConstructor(group=5, fix_d=True).coeff_ring()
Real Field with 53 bits of precision
sage: MFSeriesConstructor(group=5).coeff_ring()
Fraction Field of Univariate Polynomial Ring in d over Integer Ring
"""
return self._coeff_ring
def qseries_ring(self):
r"""
Return the used PowerSeriesRing.
EXAMPLES::
sage: MFSeriesConstructor(group=4, fix_d=True).qseries_ring()
Power Series Ring in q over Rational Field
sage: MFSeriesConstructor(group=4).qseries_ring()
Power Series Ring in q over Fraction Field of Univariate Polynomial Ring in d over Integer Ring
sage: MFSeriesConstructor(group=5, fix_d=True).qseries_ring()
Power Series Ring in q over Real Field with 53 bits of precision
sage: MFSeriesConstructor(group=5).qseries_ring()
Power Series Ring in q over Fraction Field of Univariate Polynomial Ring in d over Integer Ring
"""
return self._qseries_ring
@cached_method
def J_inv_ZZ(self):
r"""
Return the rational Fourier expansion of ``J_inv``,
where ``d`` is replaced by ``1``.
This is the main function used to determine all Fourier expansions!
EXAMPLES::
sage: MFSeriesConstructor(prec=3).J_inv_ZZ()
q^-1 + 31/72 + 1823/27648*q + O(q^2)
sage: MFSeriesConstructor(group=5, prec=3, fix_d=True).J_inv_ZZ()
q^-1 + 79/200 + 42877/640000*q + O(q^2)
sage: MFSeriesConstructor(group=5, prec=3, fix_d=True).J_inv_ZZ().parent()
Laurent Series Ring in q over Rational Field
"""
F1 = lambda a,b: self._ZZseries_ring(\
[ ZZ(0) ] + [\
rising_factorial(a,k) * rising_factorial(b,k) / (ZZ(k).factorial())**2 * sum([\
ZZ(1)/(a+j)+ZZ(1)/(b+j)-ZZ(2)/ZZ(1+j) for j in range(ZZ(0),ZZ(k))\
]) for k in range(ZZ(1),ZZ(self._prec+1))
], ZZ(self._prec+1)\
)
F = lambda a,b,c: self._ZZseries_ring([\
rising_factorial(a,k) * rising_factorial(b,k) / rising_factorial(c,k) / (ZZ(k).factorial())\
for k in range(ZZ(0),ZZ(self._prec+1))\
], ZZ(self._prec+1))
a = self._group.alpha()
b = self._group.beta()
Phi = F1(a,b) / F(a,b,ZZ(1))
q = self._ZZseries_ring.gen()
J_inv_ZZ = ZZ(1) / ((q*Phi.exp()).reversion())
return J_inv_ZZ
@cached_method
def J_inv(self):
r"""
Return the Fourier expansion of ``J_inv``.
EXAMPLES::
sage: MFSeriesConstructor(prec=3, fix_d=True).J_inv()
1/1728*q^-1 + 31/72 + 1823/16*q + O(q^2)
sage: MFSeriesConstructor(prec=3).J_inv_ZZ() == MFSeriesConstructor(prec=3, set_d=1).J_inv()
True
sage: MFSeriesConstructor(group=5, prec=3).J_inv()
d*q^-1 + 79/200 + 42877/(640000*d)*q + O(q^2)
sage: MFSeriesConstructor(group=5, prec=3, fix_d=True).J_inv()
0.00705223418128563*q^-1 + 0.395000000000000 + 9.49987064777062*q + O(q^2)
sage: MFSeriesConstructor(group=5, prec=3).J_inv().parent()
Laurent Series Ring in q over Fraction Field of Univariate Polynomial Ring in d over Integer Ring
sage: MFSeriesConstructor(group=5, prec=3, fix_d=True).J_inv().parent()
Laurent Series Ring in q over Real Field with 53 bits of precision
"""
return self.J_inv_ZZ()(self._qseries_ring.gen()/self._d)
@cached_method
def F_rho_ZZ(self):
r"""
Return the rational Fourier expansion of ``F_rho``,
where ``d`` is replaced by ``1``.
EXAMPLES::
sage: MFSeriesConstructor(prec=3).F_rho_ZZ()
1 + 5/36*q + 5/6912*q^2 + O(q^3)
sage: MFSeriesConstructor(group=5, prec=3, fix_d=True).F_rho_ZZ()
1 + 7/100*q + 21/160000*q^2 + O(q^3)
sage: MFSeriesConstructor(group=5, prec=3, fix_d=True).F_rho_ZZ().parent()
Power Series Ring in q over Rational Field
"""
q = self._ZZseries_ring.gen()
n = self.hecke_n()
temp_expr = ((-q*self.J_inv_ZZ().derivative())**2/(self.J_inv_ZZ()*(self.J_inv_ZZ()-1))).power_series()
F_rho_ZZ = (temp_expr.log()/(n-2)).exp()
return F_rho_ZZ
@cached_method
def F_rho(self):
r"""
Return the Fourier expansion of ``F_rho``.
EXAMPLES::
sage: MFSeriesConstructor(prec=3, fix_d=True).F_rho()
1 + 240*q + 2160*q^2 + O(q^3)
sage: MFSeriesConstructor(prec=3).F_rho_ZZ() == MFSeriesConstructor(prec=3, set_d=1).F_rho()
True
sage: MFSeriesConstructor(group=5, prec=3).F_rho()
1 + 7/(100*d)*q + 21/(160000*d^2)*q^2 + O(q^3)
sage: MFSeriesConstructor(group=5, prec=3, fix_d=True).F_rho()
1.00000000000000 + 9.92593243510795*q + 2.63903932249093*q^2 + O(q^3)
sage: MFSeriesConstructor(group=5, prec=3).F_rho().parent()
Power Series Ring in q over Fraction Field of Univariate Polynomial Ring in d over Integer Ring
sage: MFSeriesConstructor(group=5, prec=3, fix_d=True).F_rho().parent()
Power Series Ring in q over Real Field with 53 bits of precision
"""
return self.F_rho_ZZ()(self._qseries_ring.gen()/self._d)
@cached_method
def F_i_ZZ(self):
r"""
Return the rational Fourier expansion of ``F_i``,
where ``d`` is replaced by ``1``.
EXAMPLES::
sage: MFSeriesConstructor(prec=3).F_i_ZZ()
1 - 7/24*q - 77/13824*q^2 + O(q^3)
sage: MFSeriesConstructor(group=5, prec=3, fix_d=True).F_i_ZZ()
1 - 13/40*q - 351/64000*q^2 + O(q^3)
sage: MFSeriesConstructor(group=5, prec=3, fix_d=True).F_i_ZZ().parent()
Power Series Ring in q over Rational Field
"""
q = self._ZZseries_ring.gen()
n = self.hecke_n()
temp_expr = ((-q*self.J_inv_ZZ().derivative())**n/(self.J_inv_ZZ()**(n-1)*(self.J_inv_ZZ()-1))).power_series()
F_i_ZZ = (temp_expr.log()/(n-2)).exp()
return F_i_ZZ
@cached_method
def F_i(self):
r"""
Return the Fourier expansion of ``F_i``.
EXAMPLES::
sage: MFSeriesConstructor(prec=3, fix_d=True).F_i()
1 - 504*q - 16632*q^2 + O(q^3)
sage: MFSeriesConstructor(prec=3).F_i_ZZ() == MFSeriesConstructor(prec=3, set_d=1).F_i()
True
sage: MFSeriesConstructor(group=5, prec=3).F_i()
1 - 13/(40*d)*q - 351/(64000*d^2)*q^2 + O(q^3)
sage: MFSeriesConstructor(group=5, prec=3, fix_d=True).F_i()
1.00000000000000 - 46.0846863058583*q - 110.274143118371*q^2 + O(q^3)
sage: MFSeriesConstructor(group=5, prec=3).F_i().parent()
Power Series Ring in q over Fraction Field of Univariate Polynomial Ring in d over Integer Ring
sage: MFSeriesConstructor(group=5, prec=3, fix_d=True).F_i().parent()
Power Series Ring in q over Real Field with 53 bits of precision
"""
return self.F_i_ZZ()(self._qseries_ring.gen()/self._d)
@cached_method
def F_inf_ZZ(self):
r"""
Return the rational Fourier expansion of ``F_inf``,
where ``d`` is replaced by ``1``.
EXAMPLES::
sage: MFSeriesConstructor(prec=3).F_inf_ZZ()
q - 1/72*q^2 + 7/82944*q^3 + O(q^4)
sage: MFSeriesConstructor(group=5, prec=3, fix_d=True).F_inf_ZZ()
q - 9/200*q^2 + 279/640000*q^3 + O(q^4)
sage: MFSeriesConstructor(group=5, prec=3, fix_d=True).F_inf_ZZ().parent()
Power Series Ring in q over Rational Field
"""
q = self._ZZseries_ring.gen()
n = self.hecke_n()
temp_expr = ((-q*self.J_inv_ZZ().derivative())**(2*n)/(self.J_inv_ZZ()**(2*n-2)*(self.J_inv_ZZ()-1)**n)/q**(n-2)).power_series()
F_inf_ZZ = (temp_expr.log()/(n-2)).exp()*q
return F_inf_ZZ
@cached_method
def F_inf(self):
r"""
Return the Fourier expansion of ``F_inf``.
EXAMPLES::
sage: MFSeriesConstructor(prec=3, fix_d=True).F_inf()
q - 24*q^2 + 252*q^3 + O(q^4)
sage: MFSeriesConstructor(prec=3).F_inf_ZZ() == MFSeriesConstructor(prec=3, set_d=1).F_inf()
True
sage: MFSeriesConstructor(group=5, prec=3).F_inf()
q - 9/(200*d)*q^2 + 279/(640000*d^2)*q^3 + O(q^4)
sage: MFSeriesConstructor(group=5, prec=3, fix_d=True).F_inf()
0.000000000000000 + 1.00000000000000*q - 6.38095656542654*q^2 + 8.76538060684488*q^3 + O(q^4)
sage: MFSeriesConstructor(group=5, prec=3).F_inf().parent()
Power Series Ring in q over Fraction Field of Univariate Polynomial Ring in d over Integer Ring
sage: MFSeriesConstructor(group=5, prec=3, fix_d=True).F_inf().parent()
Power Series Ring in q over Real Field with 53 bits of precision
"""
return self._d*self.F_inf_ZZ()(self._qseries_ring.gen()/self._d)
@cached_method
def G_inv_ZZ(self):
r"""
Return the rational Fourier expansion of ``G_inv``,
where ``d`` is replaced by ``1``.
EXAMPLES::
sage: MFSeriesConstructor(group=4, prec=3).G_inv_ZZ()
q^-1 - 3/32 - 955/16384*q + O(q^2)
sage: MFSeriesConstructor(group=8, prec=3, fix_d=True).G_inv_ZZ()
q^-1 - 15/128 - 15139/262144*q + O(q^2)
sage: MFSeriesConstructor(group=8, prec=3, fix_d=True).G_inv_ZZ().parent()
Laurent Series Ring in q over Rational Field
"""
n = self.hecke_n()
if (ZZ(2).divides(n)):
return self.F_i_ZZ()*(self.F_rho_ZZ()**(ZZ(n/ZZ(2))))/self.F_inf_ZZ()
else:
#return self._qseries_ring([])
raise Exception("G_inv doesn't exist for n={}.".format(self.hecke_n()))
@cached_method
def G_inv(self):
r"""
Return the Fourier expansion of ``G_inv``.
EXAMPLES::
sage: MFSeriesConstructor(group=4, prec=3, fix_d=True).G_inv()
1/16777216*q^-1 - 3/2097152 - 955/4194304*q + O(q^2)
sage: MFSeriesConstructor(group=4, prec=3).G_inv_ZZ() == MFSeriesConstructor(group=4, prec=3, set_d=1).G_inv()
True
sage: MFSeriesConstructor(group=8, prec=3).G_inv()
d^3*q^-1 - 15*d^2/128 - 15139*d/262144*q + O(q^2)
sage: MFSeriesConstructor(group=8, prec=3, fix_d=True).G_inv()
1.64838830030189e-6*q^-1 - 0.0000163526310530017 - 0.000682197999433738*q + O(q^2)
sage: MFSeriesConstructor(group=8, prec=3).G_inv().parent()
Laurent Series Ring in q over Fraction Field of Univariate Polynomial Ring in d over Integer Ring
sage: MFSeriesConstructor(group=8, prec=3, fix_d=True).G_inv().parent()
Laurent Series Ring in q over Real Field with 53 bits of precision
"""
return (self._d)**2*self.G_inv_ZZ()(self._qseries_ring.gen()/self._d)
@cached_method
def E4_ZZ(self):
r"""
Return the rational Fourier expansion of ``E_4``,
where ``d`` is replaced by ``1``.
EXAMPLES::
sage: MFSeriesConstructor(prec=3).E4_ZZ()
1 + 5/36*q + 5/6912*q^2 + O(q^3)
sage: MFSeriesConstructor(group=5, prec=3, fix_d=True).E4_ZZ()
1 + 21/100*q + 483/32000*q^2 + O(q^3)
sage: MFSeriesConstructor(group=5, prec=3, fix_d=True).E4_ZZ().parent()
Power Series Ring in q over Rational Field
"""
q = self._ZZseries_ring.gen()
E4_ZZ = ((-q*self.J_inv_ZZ().derivative())**2/(self.J_inv_ZZ()*(self.J_inv_ZZ()-1))).power_series()
return E4_ZZ
@cached_method
def E4(self):
r"""
Return the Fourier expansion of ``E_4``.
EXAMPLES::
sage: MFSeriesConstructor(prec=3, fix_d=True).E4()
1 + 240*q + 2160*q^2 + O(q^3)
sage: MFSeriesConstructor(prec=3).E4_ZZ() == MFSeriesConstructor(prec=3, set_d=1).E4()
True
sage: MFSeriesConstructor(group=5, prec=3).E4()
1 + 21/(100*d)*q + 483/(32000*d^2)*q^2 + O(q^3)
sage: MFSeriesConstructor(group=5, prec=3, fix_d=True).E4()
1.00000000000000 + 29.7777973053239*q + 303.489522086457*q^2 + O(q^3)
sage: MFSeriesConstructor(group=5, prec=3).E4().parent()
Power Series Ring in q over Fraction Field of Univariate Polynomial Ring in d over Integer Ring
sage: MFSeriesConstructor(group=5, prec=3, fix_d=True).E4().parent()
Power Series Ring in q over Real Field with 53 bits of precision
"""
return self.E4_ZZ()(self._qseries_ring.gen()/self._d)
@cached_method
def E6_ZZ(self):
r"""
Return the rational Fourier expansion of ``E_6``,
where ``d`` is replaced by ``1``.
EXAMPLES::
sage: MFSeriesConstructor(prec=3).E6_ZZ()
1 - 7/24*q - 77/13824*q^2 + O(q^3)
sage: MFSeriesConstructor(group=5, prec=3, fix_d=True).E6_ZZ()
1 - 37/200*q - 14663/320000*q^2 + O(q^3)
sage: MFSeriesConstructor(group=5, prec=3, fix_d=True).E6_ZZ().parent()
Power Series Ring in q over Rational Field
"""
q = self._ZZseries_ring.gen()
n = self.hecke_n()
E6_ZZ = ((-q*self.J_inv_ZZ().derivative())**3/(self.J_inv_ZZ()**2*(self.J_inv_ZZ()-1))).power_series()
return E6_ZZ
@cached_method
def E6(self):
r"""
Return the Fourier expansion of ``E_6``.
EXAMPLES::
sage: MFSeriesConstructor(prec=3, fix_d=True).E6()
1 - 504*q - 16632*q^2 + O(q^3)
sage: MFSeriesConstructor(prec=3).E6_ZZ() == MFSeriesConstructor(prec=3, set_d=1).E6()
True
sage: MFSeriesConstructor(group=5, prec=3).E6()
1 - 37/(200*d)*q - 14663/(320000*d^2)*q^2 + O(q^3)
sage: MFSeriesConstructor(group=5, prec=3, fix_d=True).E6()
1.00000000000000 - 26.2328214356424*q - 921.338894897250*q^2 + O(q^3)
sage: MFSeriesConstructor(group=5, prec=3).E6().parent()
Power Series Ring in q over Fraction Field of Univariate Polynomial Ring in d over Integer Ring
sage: MFSeriesConstructor(group=5, prec=3, fix_d=True).E6().parent()
Power Series Ring in q over Real Field with 53 bits of precision
"""
return self.E6_ZZ()(self._qseries_ring.gen()/self._d)
@cached_method
def Delta_ZZ(self):
r"""
Return the rational Fourier expansion of ``Delta``,
where ``d`` is replaced by ``1``.
EXAMPLES::
sage: MFSeriesConstructor(prec=3).Delta_ZZ()
q - 1/72*q^2 + O(q^3)
sage: MFSeriesConstructor(group=5, prec=3, fix_d=True).Delta_ZZ()
71/50*q + 28267/16000*q^2 + O(q^3)
sage: MFSeriesConstructor(group=5, prec=3, fix_d=True).Delta_ZZ().parent()
Power Series Ring in q over Rational Field
"""
n = self.hecke_n()
return self.E4_ZZ()**(2*n-6)*(self.E4_ZZ()**n-self.E6_ZZ()**2)
@cached_method
def Delta(self):
r"""
Return the Fourier expansion of ``Delta``.
EXAMPLES::
sage: MFSeriesConstructor(prec=3, fix_d=True).Delta()
q - 24*q^2 + O(q^3)
sage: MFSeriesConstructor(prec=3).Delta_ZZ() == MFSeriesConstructor(prec=3, set_d=1).Delta()
True
sage: MFSeriesConstructor(group=5, prec=3).Delta()
71/50*q + 28267/(16000*d)*q^2 + O(q^3)
sage: MFSeriesConstructor(group=5, prec=3, fix_d=True).Delta()
0.000000000000000 + 1.42000000000000*q + 250.514582270711*q^2 + O(q^3)
sage: MFSeriesConstructor(group=5, prec=3).Delta().parent()
Power Series Ring in q over Fraction Field of Univariate Polynomial Ring in d over Integer Ring
sage: MFSeriesConstructor(group=5, prec=3, fix_d=True).Delta().parent()
Power Series Ring in q over Real Field with 53 bits of precision
"""
return (self._d)*self.Delta_ZZ()(self._qseries_ring.gen()/self._d)
@cached_method
def E2_ZZ(self):
r"""
Return the rational Fourier expansion of ``E2``,
where ``d`` is replaced by ``1``.
EXAMPLES::
sage: MFSeriesConstructor(prec=3).E2_ZZ()
1 - 1/72*q - 1/41472*q^2 + O(q^3)
sage: MFSeriesConstructor(group=5, prec=3, fix_d=True).E2_ZZ()
1 - 9/200*q - 369/320000*q^2 + O(q^3)
sage: MFSeriesConstructor(group=5, prec=3, fix_d=True).E2_ZZ().parent()
Power Series Ring in q over Rational Field
"""
q = self._ZZseries_ring.gen()
E2_ZZ = (q*self.F_inf_ZZ().derivative())/self.F_inf_ZZ()
return E2_ZZ
@cached_method
def E2(self):
r"""
Return the Fourier expansion of ``E2``.
EXAMPLES::
sage: MFSeriesConstructor(prec=3, fix_d=True).E2()
1 - 24*q - 72*q^2 + O(q^3)
sage: MFSeriesConstructor(prec=3).E2_ZZ() == MFSeriesConstructor(prec=3, set_d=1).E2()
True
sage: MFSeriesConstructor(group=5, prec=3).E2()
1 - 9/(200*d)*q - 369/(320000*d^2)*q^2 + O(q^3)
sage: MFSeriesConstructor(group=5, prec=3, fix_d=True).E2()
1.00000000000000 - 6.38095656542654*q - 23.1858454761703*q^2 + O(q^3)
sage: MFSeriesConstructor(group=5, prec=3).E2().parent()
Power Series Ring in q over Fraction Field of Univariate Polynomial Ring in d over Integer Ring
sage: MFSeriesConstructor(group=5, prec=3, fix_d=True).E2().parent()
Power Series Ring in q over Real Field with 53 bits of precision
"""
return self.E2_ZZ()(self._qseries_ring.gen()/self._d)
class JFSeriesConstructor(SageObject,UniqueRepresentation):
r"""
Constructor for the Fourier expansion of some
(specific, basic) modular forms.
The constructor is used by forms elements in case
their Fourier expansion is needed or requested.
"""
@staticmethod
def __classcall__(cls, group = HeckeTriangleGroup(3), prec=ZZ(10)):
r"""
Return a (cached) instance with canonical parameters.
.. NOTE:
For each choice of group and precision the constructor is
cached (only) once. Further calculations with different
base rings and possibly numerical parameters are based on
the same cached instance.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.series_constructor import JFSeriesConstructor
sage: JFSeriesConstructor() == JFSeriesConstructor(3, 10)
True
sage: JFSeriesConstructor(group=4).hecke_n()
4
sage: JFSeriesConstructor(group=5, prec=12).prec()
12
"""
if (group==infinity):
group = HeckeTriangleGroup(infinity)
else:
try:
group = HeckeTriangleGroup(ZZ(group))
except TypeError:
group = HeckeTriangleGroup(group.n())
prec=ZZ(prec)
# We don't need this assumption the precision may in principle also be negative.
# if (prec<1):
# raise Exception("prec must be an Integer >=1")
return super(JFSeriesConstructor,cls).__classcall__(cls, group, prec)
def __init__(self, group, prec):
r"""
Constructor for the Fourier expansion of some
(specific, basic) modular forms.
INPUT:
- ``group`` -- A Hecke triangle group (default: HeckeTriangleGroup(3)).
- ``prec`` -- An integer (default: 10), the default precision used
in calculations in the LaurentSeriesRing or PowerSeriesRing.
OUTPUT:
The constructor for Fourier expansion with the specified settings.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.series_constructor import JFSeriesConstructor
sage: JFC = JFSeriesConstructor()
sage: JFC
Power series constructor for Hecke modular forms for n=3 with (basic series) precision 10
sage: JFC.group()
Hecke triangle group for n = 3
sage: JFC.prec()
10
sage: JFC._series_ring
Power Series Ring in q over Rational Field
sage: JFSeriesConstructor(group=4)
Power series constructor for Hecke modular forms for n=4 with (basic series) precision 10
sage: JFSeriesConstructor(group=5, prec=12)
Power series constructor for Hecke modular forms for n=5 with (basic series) precision 12
sage: JFSeriesConstructor(group=infinity)
Power series constructor for Hecke modular forms for n=+Infinity with (basic series) precision 10
"""
self._group = group
self._prec = prec
FR = FractionField(PolynomialRing(ZZ,'p,d'))
self._series_ring = PowerSeriesRing(FR,'q',default_prec=self._prec)
#self._series_ring2 = PowerSeriesRing(QQ[['q']],'p',default_prec=self._prec)
self._qseries_ring = PowerSeriesRing(QQ,'q',default_prec=self._prec)
def _repr_(self):
r"""
Return the string representation of ``self``.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.series_constructor import JFSeriesConstructor
sage: JFSeriesConstructor(group=4)
Power series constructor for Hecke modular forms for n=4 with (basic series) precision 10
sage: JFSeriesConstructor(group=5, prec=12)
Power series constructor for Hecke modular forms for n=5 with (basic series) precision 12
"""
return "Power series constructor for Hecke modular forms for n={} with (basic series) precision {}".\
format(self._group.n(), self._prec)
def group(self):
r"""
Return the (Hecke triangle) group of ``self``.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.series_constructor import JFSeriesConstructor
sage: JFSeriesConstructor(group=4).group()
Hecke triangle group for n = 4
"""
return self._group
def hecke_n(self):
r"""
Return the parameter ``n`` of the (Hecke triangle) group of ``self``.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.series_constructor import JFSeriesConstructor
sage: JFSeriesConstructor(group=4).hecke_n()
4
"""
return self._group.n()
def prec(self):
r"""
Return the used default precision for the PowerSeriesRing or LaurentSeriesRing.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.series_constructor import JFSeriesConstructor
sage: JFSeriesConstructor(group=5).prec()
10
sage: JFSeriesConstructor(group=5, prec=20).prec()
20
"""
return self._prec
@cached_method
def J_inv_ZZ(self):
r"""
Return the rational Fourier expansion of ``J_inv``,
where the parameter ``d`` is replaced by ``1``.
This is the main function used to determine all Fourier expansions!
.. NOTE:
The Fourier expansion of ``J_inv`` for ``d!=1``
is given by ``J_inv_ZZ(q/d)``.
.. TODO:
The functions that are used in this implementation are
products of hypergeometric series with other, elementary,
functions. Implement them and clean up this representation.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.series_constructor import JFSeriesConstructor
sage: JFSeriesConstructor(prec=3).J_inv_ZZ()
q^-1 + 31/72 + 1823/27648*q + O(q^2)
sage: JFSeriesConstructor(group=5, prec=3).J_inv_ZZ()
q^-1 + 79/200 + 42877/640000*q + O(q^2)
sage: JFSeriesConstructor(group=5, prec=3).J_inv_ZZ().parent()
Laurent Series Ring in q over Rational Field
sage: JFSeriesConstructor(group=infinity, prec=3).J_inv_ZZ()
q^-1 + 3/8 + 69/1024*q + O(q^2)
"""
F1 = lambda a,b: self._qseries_ring(
[ ZZ(0) ]
+ [
rising_factorial(a,k) * rising_factorial(b,k) / (ZZ(k).factorial())**2
* sum(ZZ(1)/(a+j) + ZZ(1)/(b+j) - ZZ(2)/ZZ(1+j)
for j in range(ZZ(0),ZZ(k))
)
for k in range(ZZ(1), ZZ(self._prec+1))
],
ZZ(self._prec+1)
)
F = lambda a,b,c: self._qseries_ring(
[
rising_factorial(a,k) * rising_factorial(b,k) / rising_factorial(c,k) / ZZ(k).factorial()
for k in range(ZZ(0), ZZ(self._prec+1))
],
ZZ(self._prec+1)
)
a = self._group.alpha()
b = self._group.beta()
Phi = F1(a,b) / F(a,b,ZZ(1))
q = self._qseries_ring.gen()
# the current implementation of power series reversion is slow
# J_inv_ZZ = ZZ(1) / ((q*Phi.exp()).reversion())
temp_f = (q*Phi.exp()).polynomial()
new_f = temp_f.revert_series(temp_f.degree()+1)
J_inv_ZZ = ZZ(1) / (new_f + O(q**(temp_f.degree()+1)))
q = self._series_ring.gen()
J_inv_ZZ = sum([J_inv_ZZ.coefficients()[m] * q**J_inv_ZZ.exponents()[m] for m in range(len(J_inv_ZZ.coefficients()))]) + O(q**J_inv_ZZ.prec())
return J_inv_ZZ
@cached_method
def f_rho_ZZ(self):
r"""
Return the rational Fourier expansion of ``f_rho``,
where the parameter ``d`` is replaced by ``1``.
.. NOTE:
The Fourier expansion of ``f_rho`` for ``d!=1``
is given by ``f_rho_ZZ(q/d)``.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.series_constructor import JFSeriesConstructor
sage: JFSeriesConstructor(prec=3).f_rho_ZZ()
1 + 5/36*q + 5/6912*q^2 + O(q^3)
sage: JFSeriesConstructor(group=5, prec=3).f_rho_ZZ()
1 + 7/100*q + 21/160000*q^2 + O(q^3)
sage: JFSeriesConstructor(group=5, prec=3).f_rho_ZZ().parent()
Power Series Ring in q over Rational Field
sage: JFSeriesConstructor(group=infinity, prec=3).f_rho_ZZ()
1
"""
q = self._series_ring.gen(0)
n = self.hecke_n()
if (n == infinity):
f_rho_ZZ = self._series_ring(1)
else:
temp_expr = ((-q*self.J_inv_ZZ().derivative())**2/(self.J_inv_ZZ()*(self.J_inv_ZZ()-1))).power_series()
f_rho_ZZ = (temp_expr.log()/(n-2)).exp()
return f_rho_ZZ
@cached_method
def f_i_ZZ(self):
r"""
Return the rational Fourier expansion of ``f_i``,
where the parameter ``d`` is replaced by ``1``.
.. NOTE:
The Fourier expansion of ``f_i`` for ``d!=1``
is given by ``f_i_ZZ(q/d)``.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.series_constructor import JFSeriesConstructor
sage: JFSeriesConstructor(prec=3).f_i_ZZ()
1 - 7/24*q - 77/13824*q^2 + O(q^3)
sage: JFSeriesConstructor(group=5, prec=3).f_i_ZZ()
1 - 13/40*q - 351/64000*q^2 + O(q^3)
sage: JFSeriesConstructor(group=5, prec=3).f_i_ZZ().parent()
Power Series Ring in q over Rational Field
sage: JFSeriesConstructor(group=infinity, prec=3).f_i_ZZ()
1 - 3/8*q + 3/512*q^2 + O(q^3)
"""
q = self._series_ring.gen(0)
n = self.hecke_n()
if (n == infinity):
f_i_ZZ = (-q*self.J_inv_ZZ().derivative()/self.J_inv_ZZ()).power_series()
else:
temp_expr = ((-q*self.J_inv_ZZ().derivative())**n/(self.J_inv_ZZ()**(n-1)*(self.J_inv_ZZ()-1))).power_series()
f_i_ZZ = (temp_expr.log()/(n-2)).exp()
return f_i_ZZ
@cached_method
def f_inf_ZZ(self):
r"""
Return the rational Fourier expansion of ``f_inf``,
where the parameter ``d`` is replaced by ``1``.
.. NOTE:
The Fourier expansion of ``f_inf`` for ``d!=1``
is given by ``d*f_inf_ZZ(q/d)``.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.series_constructor import JFSeriesConstructor
sage: JFSeriesConstructor(prec=3).f_inf_ZZ()
q - 1/72*q^2 + 7/82944*q^3 + O(q^4)
sage: JFSeriesConstructor(group=5, prec=3).f_inf_ZZ()
q - 9/200*q^2 + 279/640000*q^3 + O(q^4)
sage: JFSeriesConstructor(group=5, prec=3).f_inf_ZZ().parent()
Power Series Ring in q over Rational Field
sage: JFSeriesConstructor(group=infinity, prec=3).f_inf_ZZ()
q - 1/8*q^2 + 7/1024*q^3 + O(q^4)
"""
q = self._series_ring.gen(0)
n = self.hecke_n()
if (n == infinity):
f_inf_ZZ = ((-q*self.J_inv_ZZ().derivative())**2/(self.J_inv_ZZ()**2*(self.J_inv_ZZ()-1))).power_series()
else:
temp_expr = ((-q*self.J_inv_ZZ().derivative())**(2*n)/(self.J_inv_ZZ()**(2*n-2)*(self.J_inv_ZZ()-1)**n)/q**(n-2)).power_series()
f_inf_ZZ = (temp_expr.log()/(n-2)).exp()*q
return f_inf_ZZ
@cached_method
def G_inv_ZZ(self):
r"""
Return the rational Fourier expansion of ``G_inv``,
where the parameter ``d`` is replaced by ``1``.
.. NOTE:
The Fourier expansion of ``G_inv`` for ``d!=1``
is given by ``d*G_inv_ZZ(q/d)``.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.series_constructor import JFSeriesConstructor
sage: JFSeriesConstructor(group=4, prec=3).G_inv_ZZ()
q^-1 - 3/32 - 955/16384*q + O(q^2)
sage: JFSeriesConstructor(group=8, prec=3).G_inv_ZZ()
q^-1 - 15/128 - 15139/262144*q + O(q^2)
sage: JFSeriesConstructor(group=8, prec=3).G_inv_ZZ().parent()
Laurent Series Ring in q over Rational Field
sage: JFSeriesConstructor(group=infinity, prec=3).G_inv_ZZ()
q^-1 - 1/8 - 59/1024*q + O(q^2)
"""
n = self.hecke_n()
# Note that G_inv is not a weakly holomorphic form (because of the behavior at -1)
if (n == infinity):
q = self._series_ring.gen(0)
temp_expr = (self.J_inv_ZZ()/self.f_inf_ZZ()*q**2).power_series()
return 1/q*self.f_i_ZZ()*(temp_expr.log()/2).exp()
elif (ZZ(2).divides(n)):
return self.f_i_ZZ()*(self.f_rho_ZZ()**(ZZ(n/ZZ(2))))/self.f_inf_ZZ()
else:
#return self._qseries_ring([])
raise ValueError("G_inv doesn't exist for n={}.".format(self.hecke_n()))
@cached_method
def E4_ZZ(self):
r"""
Return the rational Fourier expansion of ``E_4``,
where the parameter ``d`` is replaced by ``1``.
.. NOTE:
The Fourier expansion of ``E4`` for ``d!=1``
is given by ``E4_ZZ(q/d)``.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.series_constructor import JFSeriesConstructor
sage: JFSeriesConstructor(prec=3).E4_ZZ()
1 + 5/36*q + 5/6912*q^2 + O(q^3)
sage: JFSeriesConstructor(group=5, prec=3).E4_ZZ()
1 + 21/100*q + 483/32000*q^2 + O(q^3)
sage: JFSeriesConstructor(group=5, prec=3).E4_ZZ().parent()
Power Series Ring in q over Rational Field
sage: JFSeriesConstructor(group=infinity, prec=3).E4_ZZ()
1 + 1/4*q + 7/256*q^2 + O(q^3)
"""
q = self._series_ring.gen(0)
E4_ZZ = ((-q*self.J_inv_ZZ().derivative())**2/(self.J_inv_ZZ()*(self.J_inv_ZZ()-1))).power_series()
return E4_ZZ
@cached_method
def E6_ZZ(self):
r"""
Return the rational Fourier expansion of ``E_6``,
where the parameter ``d`` is replaced by ``1``.
.. NOTE:
The Fourier expansion of ``E6`` for ``d!=1``
is given by ``E6_ZZ(q/d)``.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.series_constructor import JFSeriesConstructor
sage: JFSeriesConstructor(prec=3).E6_ZZ()
1 - 7/24*q - 77/13824*q^2 + O(q^3)
sage: JFSeriesConstructor(group=5, prec=3).E6_ZZ()
1 - 37/200*q - 14663/320000*q^2 + O(q^3)
sage: JFSeriesConstructor(group=5, prec=3).E6_ZZ().parent()
Power Series Ring in q over Rational Field
sage: JFSeriesConstructor(group=infinity, prec=3).E6_ZZ()
1 - 1/8*q - 31/512*q^2 + O(q^3)
"""
q = self._series_ring.gen(0)
E6_ZZ = ((-q*self.J_inv_ZZ().derivative())**3/(self.J_inv_ZZ()**2*(self.J_inv_ZZ()-1))).power_series()
return E6_ZZ
@cached_method
def Delta_ZZ(self):
r"""
Return the rational Fourier expansion of ``Delta``,
where the parameter ``d`` is replaced by ``1``.
.. NOTE:
The Fourier expansion of ``Delta`` for ``d!=1``
is given by ``d*Delta_ZZ(q/d)``.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.series_constructor import JFSeriesConstructor
sage: JFSeriesConstructor(prec=3).Delta_ZZ()
q - 1/72*q^2 + 7/82944*q^3 + O(q^4)
sage: JFSeriesConstructor(group=5, prec=3).Delta_ZZ()
q + 47/200*q^2 + 11367/640000*q^3 + O(q^4)
sage: JFSeriesConstructor(group=5, prec=3).Delta_ZZ().parent()
Power Series Ring in q over Rational Field
sage: JFSeriesConstructor(group=infinity, prec=3).Delta_ZZ()
q + 3/8*q^2 + 63/1024*q^3 + O(q^4)
"""
return (self.f_inf_ZZ()**3*self.J_inv_ZZ()**2/(self.f_rho_ZZ()**6)).power_series()
@cached_method
def E2_ZZ(self):
r"""
Return the rational Fourier expansion of ``E2``,
where the parameter ``d`` is replaced by ``1``.
.. NOTE:
The Fourier expansion of ``E2`` for ``d!=1``
is given by ``E2_ZZ(q/d)``.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.series_constructor import JFSeriesConstructor
sage: JFSeriesConstructor(prec=3).E2_ZZ()
1 - 1/72*q - 1/41472*q^2 + O(q^3)
sage: JFSeriesConstructor(group=5, prec=3).E2_ZZ()
1 - 9/200*q - 369/320000*q^2 + O(q^3)
sage: JFSeriesConstructor(group=5, prec=3).E2_ZZ().parent()
Power Series Ring in q over Rational Field
sage: JFSeriesConstructor(group=infinity, prec=3).E2_ZZ()
1 - 1/8*q - 1/512*q^2 + O(q^3)
"""
q = self._series_ring.gen(0)
E2_ZZ = (q*self.f_inf_ZZ().derivative())/self.f_inf_ZZ()
return E2_ZZ
@cached_method
def EisensteinSeries_ZZ(self, k):
r"""
Return the rational Fourier expansion of the normalized Eisenstein series
of weight ``k``, where the parameter ``d`` is replaced by ``1``.
Only arithmetic groups with ``n < infinity`` are supported!
.. NOTE:
THe Fourier expansion of the series is given by ``EisensteinSeries_ZZ(q/d)``.
INPUT:
- ``k`` -- A non-negative even integer, namely the weight.
EXAMPLES::
sage: from sage.modular.modform_hecketriangle.series_constructor import JFSeriesConstructor
sage: JFC = JFSeriesConstructor(prec=6)
sage: JFC.EisensteinSeries_ZZ(k=0)
1
sage: JFC.EisensteinSeries_ZZ(k=2)
1 - 1/72*q - 1/41472*q^2 - 1/53747712*q^3 - 7/371504185344*q^4 - 1/106993205379072*q^5 + O(q^6)
sage: JFC.EisensteinSeries_ZZ(k=6)
1 - 7/24*q - 77/13824*q^2 - 427/17915904*q^3 - 7399/123834728448*q^4 - 3647/35664401793024*q^5 + O(q^6)
sage: JFC.EisensteinSeries_ZZ(k=12)
1 + 455/8292*q + 310765/4776192*q^2 + 20150585/6189944832*q^3 + 1909340615/42784898678784*q^4 + 3702799555/12322050819489792*q^5 + O(q^6)
sage: JFC.EisensteinSeries_ZZ(k=12).parent()
Power Series Ring in q over Rational Field
sage: JFC = JFSeriesConstructor(group=4, prec=5)
sage: JFC.EisensteinSeries_ZZ(k=2)
1 - 1/32*q - 5/8192*q^2 - 1/524288*q^3 - 13/536870912*q^4 + O(q^5)
sage: JFC.EisensteinSeries_ZZ(k=4)
1 + 3/16*q + 39/4096*q^2 + 21/262144*q^3 + 327/268435456*q^4 + O(q^5)
sage: JFC.EisensteinSeries_ZZ(k=6)
1 - 7/32*q - 287/8192*q^2 - 427/524288*q^3 - 9247/536870912*q^4 + O(q^5)
sage: JFC.EisensteinSeries_ZZ(k=12)
1 + 63/11056*q + 133119/2830336*q^2 + 2790081/181141504*q^3 + 272631807/185488900096*q^4 + O(q^5)
sage: JFC = JFSeriesConstructor(group=6, prec=5)
sage: JFC.EisensteinSeries_ZZ(k=2)
1 - 1/18*q - 1/648*q^2 - 7/209952*q^3 - 7/22674816*q^4 + O(q^5)
sage: JFC.EisensteinSeries_ZZ(k=4)
1 + 2/9*q + 1/54*q^2 + 37/52488*q^3 + 73/5668704*q^4 + O(q^5)
sage: JFC.EisensteinSeries_ZZ(k=6)
1 - 1/6*q - 11/216*q^2 - 271/69984*q^3 - 1057/7558272*q^4 + O(q^5)
sage: JFC.EisensteinSeries_ZZ(k=12)
1 + 182/151329*q + 62153/2723922*q^2 + 16186807/882550728*q^3 + 381868123/95315478624*q^4 + O(q^5)
"""
try:
if k < 0:
raise TypeError(None)
k = 2*ZZ(k/2)
except TypeError:
raise TypeError("k={} has to be a non-negative even integer!".format(k))
if (not self.group().is_arithmetic() or self.group().n() == infinity):
# Exceptional cases should be called manually (see in FormsRing_abstract)
raise NotImplementedError("Eisenstein series are only supported in the finite arithmetic cases!")
# Trivial case
if k == 0:
return self._series_ring(1)
M = ZZ(self.group().lam()**2)
lamk = M**(ZZ(k/2))
dval = self.group().dvalue()
def coeff(m):
m = ZZ(m)
if m < 0:
return ZZ(0)
elif m == 0:
return ZZ(1)
factor = -2*k / QQ(bernoulli(k)) / lamk
sum1 = sigma(m, k-1)
if M.divides(m):
sum2 = (lamk-1) * sigma(ZZ(m/M), k-1)
else:
sum2 = ZZ(0)
if (M == 1):
sum3 = ZZ(0)
else:
if (m == 1):
N = ZZ(1)
else:
N = ZZ(m / M**ZZ(m.valuation(M)))
sum3 = -sigma(ZZ(N), k-1) * ZZ(m/N)**(k-1) / (lamk + 1)
return factor * (sum1 + sum2 + sum3) * dval**m
q = self._series_ring.gen(0)
return sum([coeff(m)*q**m for m in range(self.prec())]).add_bigoh(self.prec())
#TODO: We only need this for the classical case where the coefficients of the eisenstein series
# are known explicitely (no HeckeTriangleGroup necessary)
#TODO: Rename all methods and remove everything unnecessary
#TODO2: use a very clear convention what is meant by q,s,p,etc...
def _cstheta(self, d, r):
#returns coefficient c(d,r) p^r q^d of i q^(-1/8) theta1; s = shift = i q^{-1/8}
k = r - QQ(1/2)
if k in ZZ and 2*d == (k**2+k):
return (-1)**k
else:
return 0
@cached_method
def _cseta(self, m):
#returns coefficient q^m of q^{1/24} eta^{-1}
if m < 0: return 0
return Partitions(m).cardinality()
@cached_method
def _cseta3(self, m):
#returns coeffcient q^m of q^{1/8} eta^{-3}
if m < 0: return 0
return PartitionTuples(3,m).cardinality()
@cached_method
def _cK(self, d, r):
#K = i theta_1 / \eta^3 = stheta*seta3
#return coefficient c(d,r) p^r q^d
## Since Jacobi form coffs satisfy r^2 < 4mn. theta1 coffs satisfy r^2 < 2 n. Here n = d + 1/8 so zero except for r^2 <= 2d + 1/4.
if d < 0: return 0
return sum( self._cseta3(d - a)*self._cstheta(a,r) for a in range(0,d+1) )
@cached_method
def _cK2(self, d, r):
## theta1^2 satisfy 4^2 < 4 n. n = d + 1/4, we get c(d,r) non-zero for r^2 <= 4d + 1.
if d < 0: return 0
coff = 0
for a in range(0,d+1):
L = sqrt(2*a + QQ(1/4)).round()
#print "a,L", a, L
coff += sum( self._cK(d - a,r-QQ(b/2))*self._cK(a,QQ(b/2)) for b in range(-2*L,2*L+1) if is_odd(b) )
return coff
@cached_method
def _cWP(self, d, r):
#\wp(z) & = \frac{1}{12} + \frac{p}{(1-p)^2} + \sum_{k,r \geq 1} k (p^k - 2 + p^{-k}) q^{kr}
if d < 0: return 0
if d == 0:
if r < 0: return 0
elif r == 0: return QQ(1/12)
else: return r
if d > 0:
if r == 0: return -2*sigma(d,1)
elif d % r == 0:
return abs(r)
else:
return 0
@cached_method
def _cZ(self, d, r):
#elliptic genus 24*K**2*wp
if d < 0: return 0
coff = 0
for a in range(0,d + 1):
L = sqrt(4*a + 1).round()
coff += sum( 24*self._cWP(d - a,r-b)*self._cK2(a,b) for b in range(-L,L+1) )
return coff
@cached_method
def K(self):
q = self._series_ring.gen()
p = self._series_ring.base_ring().gen()
def min_rcoeff(d):
#TODO
return -10
def max_rcoeff(d):
#TODO
return 10
return sum([sum([self._cK(d,r)*p**r for r in range(min_rcoeff(d), max_rcoeff(d) + 1)])*q**d for d in range(self.prec())])
@cached_method
def wp(self):
q = self._series_ring.gen()
p = self._series_ring.base_ring().gen()
def min_rcoeff(d):
#TODO
return -10
def max_rcoeff(d):
#TODO
return 10
return sum([sum([self._cWP(d,r)*p**r for r in range(min_rcoeff(d), max_rcoeff(d) + 1)])*q**d for d in range(self.prec())])
#TODO
@cached_method
def J1(self):
q = self._series_ring.gen()
p = self._series_ring.base_ring().gen()
def min_rcoeff(d):
#TODO
return -10
def max_rcoeff(d):
#TODO
return 10
return sum([sum([self._cK(d,r)*p**r for r in range(min_rcoeff(d), max_rcoeff(d) + 1)])*q**d for d in range(self.prec())])
def series(self, n=2, quadratic_twist=+1, prec=5):
r"""
Returns the `n`-th approximation to the `p`-adic L-series as a
power series in `T` (corresponding to `\gamma-1` with
`\gamma=1+p` as a generator of `1+p\ZZ_p`). Each coefficient
is a `p`-adic number whose precision is provably correct.
Here the normalization of the `p`-adic L-series is chosen such
that `L_p(J,1) = (1-1/\alpha)^2 L(J,1)/\Omega_J` where
`\alpha` is the unit root
INPUT:
- ``n`` - (default: 2) a positive integer
- ``quadratic_twist`` - (default: +1) a fundamental
discriminant of a quadratic field, coprime to the
conductor of the curve
- ``prec`` - (default: 5) maximal number of terms of the
series to compute; to compute as many as possible just
give a very large number for ``prec``; the result will
still be correct.
ALIAS: power_series is identical to series.
EXAMPLES::
sage: J = J0(188)[0]
sage: p = 7
sage: L = J.padic_lseries(p)
sage: L.is_ordinary()
True
sage: f = L.series(2)
sage: f[0]
O(7^20)
sage: f[1].norm()
3 + 4*7 + 3*7^2 + 6*7^3 + 5*7^4 + 5*7^5 + 6*7^6 + 4*7^7 + 5*7^8 + 7^10 + 5*7^11 + 4*7^13 + 4*7^14 + 5*7^15 + 2*7^16 + 5*7^17 + 7^18 + 7^19 + O(7^20)
"""
n = ZZ(n)
if n < 1:
raise ValueError("n (={0}) must be a positive integer".format(n))
if not self.is_ordinary():
raise ValueError("p (={0}) must be an ordinary prime".format(
self._p))
# check if the conditions on quadratic_twist are satisfied
D = ZZ(quadratic_twist)
if D != 1:
if D % 4 == 0:
d = D // 4
if not d.is_squarefree() or d % 4 == 1:
raise ValueError(
"quadratic_twist (={0}) must be a fundamental discriminant of a quadratic field"
.format(D))
else:
if not D.is_squarefree() or D % 4 != 1:
raise ValueError(
"quadratic_twist (={0}) must be a fundamental discriminant of a quadratic field"
.format(D))
if gcd(D, self._p) != 1:
raise ValueError(
"quadratic twist (={0}) must be coprime to p (={1}) ".
format(D, self._p))
if gcd(D, self._E.conductor()) != 1:
for ell in prime_divisors(D):
if valuation(self._E.conductor(), ell) > valuation(D, ell):
raise ValueError(
"can not twist a curve of conductor (={0}) by the quadratic twist (={1})."
.format(self._E.conductor(), D))
p = self._p
if p == 2 and self._normalize:
print('Warning : For p=2 the normalization might not be correct !')
#verbose("computing L-series for p=%s, n=%s, and prec=%s"%(p,n,prec))
# bounds = self._prec_bounds(n,prec)
# padic_prec = max(bounds[1:]) + 5
padic_prec = 10
# verbose("using p-adic precision of %s"%padic_prec)
res_series_prec = min(p**(n - 1), prec)
verbose("using series precision of %s" % res_series_prec)
ans = self._get_series_from_cache(n, res_series_prec, D)
if not ans is None:
verbose("found series in cache")
return ans
K = QQ
gamma = K(1 + p)
R = PowerSeriesRing(K, 'T', res_series_prec)
T = R(R.gen(), res_series_prec)
#L = R(0)
one_plus_T_factor = R(1)
gamma_power = K(1)
teich = self.teichmuller(padic_prec)
p_power = p**(n - 1)
# F = Qp(p,padic_prec)
verbose("Now iterating over %s summands" % ((p - 1) * p_power))
verbose_level = get_verbose()
count_verb = 0
alphas = self.alpha()
#print len(alphas)
Lprod = []
self._emb = 0
if len(alphas) == 2:
split = True
else:
split = False
for alpha in alphas:
L = R(0)
self._emb = self._emb + 1
for j in range(p_power):
s = K(0)
if verbose_level >= 2 and j / p_power * 100 > count_verb + 3:
verbose("%.2f percent done" % (float(j) / p_power * 100))
count_verb += 3
for a in range(1, p):
if split:
b = (teich[a]) % ZZ(p**n)
b = b * gamma_power
else:
b = teich[a] * gamma_power
s += self.measure(b, n, padic_prec, D, alpha)
L += s * one_plus_T_factor
one_plus_T_factor *= 1 + T
gamma_power *= gamma
Lprod = Lprod + [L]
if len(Lprod) == 1:
return Lprod[0]
else:
return Lprod[0] * Lprod[1]
def double_integral_zero_infty(Phi, tau1, tau2):
p = Phi.parent().prime()
K = tau1.parent()
R = PolynomialRing(K, 'x')
x = R.gen()
R1 = PowerSeriesRing(K, 'r1')
r1 = R1.gen()
try:
R1.set_default_prec(Phi.precision_absolute())
except AttributeError:
R1.set_default_prec(Phi.precision_relative())
level = Phi._map._manin.level()
E0inf = [M2Z([0, -1, level, 0])]
E0Zp = [M2Z([p, a, 0, 1]) for a in range(p)]
predicted_evals = num_evals(tau1, tau2)
a, b, c, d = find_center(p, level, tau1, tau2).list()
h = M2Z([a, b, c, d])
E = [h * e0 for e0 in E0Zp + E0inf]
resadd = 0
resmul = 1
total_evals = 0
percentage = QQ(0)
ii = 0
f = (x - tau2) / (x - tau1)
while len(E) > 0:
ii += 1
increment = QQ((100 - percentage) / len(E))
verbose(
'remaining %s percent (and done %s of %s evaluations)' %
(RealField(10)(100 - percentage), total_evals, predicted_evals))
newE = []
for e in E:
a, b, c, d = e.list()
assert ZZ(c) % level == 0
try:
y0 = f((a * r1 + b) / (c * r1 + d))
val = y0(y0.parent().base_ring()(0))
if all([xx.valuation(p) > 0 for xx in (y0 / val - 1).list()]):
if total_evals % 100 == 0:
Phi._map._codomain.clear_cache()
pol = val.log(p_branch=0) + (
(y0.derivative() / y0).integral())
V = [0] * pol.valuation() + pol.shift(
-pol.valuation()).list()
try:
phimap = Phi._map(M2Z([b, d, a, c]))
except OverflowError:
print(a, b, c, d)
raise OverflowError, 'Matrix too large?'
# mu_e0 = ZZ(phimap.moment(0).rational_reconstruction())
mu_e0 = ZZ(Phi._liftee._map(M2Z([b, d, a, c])).moment(0))
mu_e = [mu_e0] + [
phimap.moment(o).lift() for o in range(1, len(V))
]
resadd += sum(starmap(mul, izip(V, mu_e)))
resmul *= val**mu_e0
percentage += increment
total_evals += 1
else:
newE.extend([e * e0 for e0 in E0Zp])
except ZeroDivisionError:
#raise RuntimeError,'Probably not enough working precision...'
newE.extend([e * e0 for e0 in E0Zp])
E = newE
verbose('total evaluations = %s' % total_evals)
val = resmul.valuation()
return p**val * K.teichmuller(p**(-val) * resmul) * resadd.exp()
