def _element_constructor_(self, x, e=1):
r"""Construct a Puiseux series from `x`.
INPUT:
- ``x`` -- an object that can be converted into a Puiseux series in (x-a)
- ``a`` -- (default: 0) the series is in powers of (var - a)
- ``e`` -- (default: 1) the ramification index of the series
"""
P = parent(x)
# 1. x is a Puiseux series belonging to this ring
if isinstance(x, self.element_class) and P is self:
return x
# 2. x is a Puiseux series but not an element of this ring. the laurent
# part should be coercible to the laurent series ring of self
elif isinstance(x, self.element_class):
l = self.laurent_series_ring()(x.laurent_part)
e = x.ramification_index
# 3. x is a member of the base ring then convert x to a laurent series
# and set the ramificaiton index of the Puiseux series to 1.
elif P is self.base_ring():
l = self.laurent_series_ring()(x)
e = 1
# 4. x is a Laurent or power series with the same base ring
elif ((is_LaurentSeries(x) or is_PowerSeries(x))
and P is self.base_ring()):
l = self.laurent_series_ring()(x)
# 5. everything else: try to coerce to laurent series ring
else:
l = self.laurent_series_ring()(x)
e = 1
return self.element_class(self, l, e=e)
def _element_constructor_(self, x, e=1):
r"""Construct a Puiseux series from `x`.
INPUT:
- ``x`` -- an object that can be converted into a Puiseux series in (x-a)
- ``a`` -- (default: 0) the series is in powers of (var - a)
- ``e`` -- (default: 1) the ramification index of the series
"""
P = parent(x)
# 1. x is a Puiseux series belonging to this ring
if isinstance(x, self.element_class) and P is self:
return x
# 2. x is a Puiseux series but not an element of this ring. the laurent
# part should be coercible to the laurent series ring of self
elif isinstance(x, self.element_class):
l = self.laurent_series_ring()(x.laurent_part)
e = x.ramification_index
# 3. x is a member of the base ring then convert x to a laurent series
# and set the ramificaiton index of the Puiseux series to 1.
elif P is self.base_ring():
l = self.laurent_series_ring()(x)
e = 1
# 4. x is a Laurent or power series with the same base ring
elif ((is_LaurentSeries(x) or is_PowerSeries(x))
and P is self.base_ring()):
l = self.laurent_series_ring()(x)
# 5. everything else: try to coerce to laurent series ring
else:
l = self.laurent_series_ring()(x)
e = 1
return self.element_class(self, l, e=e)
def hecke_operator_on_qexp(f, n, k, eps = None,
prec=None, check=True, _return_list=False):
r"""
Given the `q`-expansion `f` of a modular form with character
`\varepsilon`, this function computes the image of `f` under the
Hecke operator `T_{n,k}` of weight `k`.
EXAMPLES::
sage: M = ModularForms(1,12)
sage: hecke_operator_on_qexp(M.basis()[0], 3, 12)
252*q - 6048*q^2 + 63504*q^3 - 370944*q^4 + O(q^5)
sage: hecke_operator_on_qexp(M.basis()[0], 1, 12, prec=7)
q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 - 6048*q^6 + O(q^7)
sage: hecke_operator_on_qexp(M.basis()[0], 1, 12)
q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 - 6048*q^6 - 16744*q^7 + 84480*q^8 - 113643*q^9 - 115920*q^10 + 534612*q^11 - 370944*q^12 - 577738*q^13 + O(q^14)
sage: M.prec(20)
20
sage: hecke_operator_on_qexp(M.basis()[0], 3, 12)
252*q - 6048*q^2 + 63504*q^3 - 370944*q^4 + 1217160*q^5 - 1524096*q^6 + O(q^7)
sage: hecke_operator_on_qexp(M.basis()[0], 1, 12)
q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 - 6048*q^6 - 16744*q^7 + 84480*q^8 - 113643*q^9 - 115920*q^10 + 534612*q^11 - 370944*q^12 - 577738*q^13 + 401856*q^14 + 1217160*q^15 + 987136*q^16 - 6905934*q^17 + 2727432*q^18 + 10661420*q^19 - 7109760*q^20 + O(q^21)
sage: (hecke_operator_on_qexp(M.basis()[0], 1, 12)*252).add_bigoh(7)
252*q - 6048*q^2 + 63504*q^3 - 370944*q^4 + 1217160*q^5 - 1524096*q^6 + O(q^7)
sage: hecke_operator_on_qexp(M.basis()[0], 6, 12)
-6048*q + 145152*q^2 - 1524096*q^3 + O(q^4)
An example on a formal power series::
sage: R.<q> = QQ[[]]
sage: f = q + q^2 + q^3 + q^7 + O(q^8)
sage: hecke_operator_on_qexp(f, 3, 12)
q + O(q^3)
sage: hecke_operator_on_qexp(delta_qexp(24), 3, 12).prec()
8
sage: hecke_operator_on_qexp(delta_qexp(25), 3, 12).prec()
9
An example of computing `T_{p,k}` in characteristic `p`::
sage: p = 199
sage: fp = delta_qexp(prec=p^2+1, K=GF(p))
sage: tfp = hecke_operator_on_qexp(fp, p, 12)
sage: tfp == fp[p] * fp
True
sage: tf = hecke_operator_on_qexp(delta_qexp(prec=p^2+1), p, 12).change_ring(GF(p))
sage: tfp == tf
True
"""
if eps is None:
# Need to have base_ring=ZZ to work over finite fields, since
# ZZ can coerce to GF(p), but QQ can't.
eps = DirichletGroup(1, base_ring=ZZ)[0]
if check:
if not (is_PowerSeries(f) or is_ModularFormElement(f)):
raise TypeError("f (=%s) must be a power series or modular form"%f)
if not is_DirichletCharacter(eps):
raise TypeError("eps (=%s) must be a Dirichlet character"%eps)
k = Integer(k)
n = Integer(n)
v = []
if prec is None:
if is_ModularFormElement(f):
# always want at least three coefficients, but not too many, unless
# requested
pr = max(f.prec(), f.parent().prec(), (n+1)*3)
pr = min(pr, 100*(n+1))
prec = pr // n + 1
else:
prec = (f.prec() / ZZ(n)).ceil()
if prec == Infinity: prec = f.parent().default_prec() // n + 1
if f.prec() < prec:
f._compute_q_expansion(prec)
p = Integer(f.base_ring().characteristic())
if k != 1 and p.is_prime() and n.is_power_of(p):
# if computing T_{p^a} in characteristic p, use the simpler (and faster)
# formula
v = [f[m*n] for m in range(prec)]
else:
l = k-1
for m in range(prec):
am = sum([eps(d) * d**l * f[m*n//(d*d)] for \
d in divisors(gcd(n, m)) if (m*n) % (d*d) == 0])
v.append(am)
if _return_list:
return v
if is_ModularFormElement(f):
R = f.parent()._q_expansion_ring()
else:
R = f.parent()
return R(v, prec)
def hecke_operator_on_basis(B, n, k, eps=None,
already_echelonized = False):
r"""
Given a basis `B` of `q`-expansions for a space of modular forms
with character `\varepsilon` to precision at least `\#B\cdot n+1`,
this function computes the matrix of `T_n` relative to `B`.
.. note::
If the elements of B are not known to sufficient precision,
this function will report that the vectors are linearly
dependent (since they are to the specified precision).
INPUT:
- ``B`` - list of q-expansions
- ``n`` - an integer >= 1
- ``k`` - an integer
- ``eps`` - Dirichlet character
- ``already_echelonized`` -- bool (default: False); if True, use that the
basis is already in Echelon form, which saves a lot of time.
EXAMPLES::
sage: sage.modular.modform.constructor.ModularForms_clear_cache()
sage: ModularForms(1,12).q_expansion_basis()
[
q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 + O(q^6),
1 + 65520/691*q + 134250480/691*q^2 + 11606736960/691*q^3 + 274945048560/691*q^4 + 3199218815520/691*q^5 + O(q^6)
]
sage: hecke_operator_on_basis(ModularForms(1,12).q_expansion_basis(), 3, 12)
Traceback (most recent call last):
...
ValueError: The given basis vectors must be linearly independent.
sage: hecke_operator_on_basis(ModularForms(1,12).q_expansion_basis(30), 3, 12)
[ 252 0]
[ 0 177148]
TESTS:
This shows that the problem with finite fields reported at :trac:`8281` is solved::
sage: bas_mod5 = [f.change_ring(GF(5)) for f in victor_miller_basis(12, 20)]
sage: hecke_operator_on_basis(bas_mod5, 2, 12)
[4 0]
[0 1]
This shows that empty input is handled sensibly (:trac:`12202`)::
sage: x = hecke_operator_on_basis([], 3, 12); x
[]
sage: x.parent()
Full MatrixSpace of 0 by 0 dense matrices over Cyclotomic Field of order 1 and degree 1
sage: y = hecke_operator_on_basis([], 3, 12, eps=DirichletGroup(13).0^2); y
[]
sage: y.parent()
Full MatrixSpace of 0 by 0 dense matrices over Cyclotomic Field of order 12 and degree 4
"""
if not isinstance(B, (list, tuple)):
raise TypeError("B (=%s) must be a list or tuple"%B)
if len(B) == 0:
if eps is None:
R = CyclotomicField(1)
else:
R = eps.base_ring()
return MatrixSpace(R, 0)(0)
f = B[0]
R = f.base_ring()
if eps is None:
eps = DirichletGroup(1, R)[0]
all_powerseries = True
for x in B:
if not is_PowerSeries(x):
all_powerseries = False
if not all_powerseries:
raise TypeError("each element of B must be a power series")
n = Integer(n)
k = Integer(k)
prec = (f.prec()-1)//n
A = R**prec
V = A.span_of_basis([g.padded_list(prec) for g in B],
already_echelonized = already_echelonized)
return _hecke_operator_on_basis(B, V, n, k, eps)
def hecke_operator_on_qexp(f,
n,
k,
eps=None,
prec=None,
check=True,
_return_list=False):
r"""
Given the `q`-expansion `f` of a modular form with character
`\varepsilon`, this function computes the image of `f` under the
Hecke operator `T_{n,k}` of weight `k`.
EXAMPLES::
sage: M = ModularForms(1,12)
sage: hecke_operator_on_qexp(M.basis()[0], 3, 12)
252*q - 6048*q^2 + 63504*q^3 - 370944*q^4 + O(q^5)
sage: hecke_operator_on_qexp(M.basis()[0], 1, 12, prec=7)
q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 - 6048*q^6 + O(q^7)
sage: hecke_operator_on_qexp(M.basis()[0], 1, 12)
q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 - 6048*q^6 - 16744*q^7 + 84480*q^8 - 113643*q^9 - 115920*q^10 + 534612*q^11 - 370944*q^12 - 577738*q^13 + O(q^14)
sage: M.prec(20)
20
sage: hecke_operator_on_qexp(M.basis()[0], 3, 12)
252*q - 6048*q^2 + 63504*q^3 - 370944*q^4 + 1217160*q^5 - 1524096*q^6 + O(q^7)
sage: hecke_operator_on_qexp(M.basis()[0], 1, 12)
q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 - 6048*q^6 - 16744*q^7 + 84480*q^8 - 113643*q^9 - 115920*q^10 + 534612*q^11 - 370944*q^12 - 577738*q^13 + 401856*q^14 + 1217160*q^15 + 987136*q^16 - 6905934*q^17 + 2727432*q^18 + 10661420*q^19 - 7109760*q^20 + O(q^21)
sage: (hecke_operator_on_qexp(M.basis()[0], 1, 12)*252).add_bigoh(7)
252*q - 6048*q^2 + 63504*q^3 - 370944*q^4 + 1217160*q^5 - 1524096*q^6 + O(q^7)
sage: hecke_operator_on_qexp(M.basis()[0], 6, 12)
-6048*q + 145152*q^2 - 1524096*q^3 + O(q^4)
An example on a formal power series::
sage: R.<q> = QQ[[]]
sage: f = q + q^2 + q^3 + q^7 + O(q^8)
sage: hecke_operator_on_qexp(f, 3, 12)
q + O(q^3)
sage: hecke_operator_on_qexp(delta_qexp(24), 3, 12).prec()
8
sage: hecke_operator_on_qexp(delta_qexp(25), 3, 12).prec()
9
An example of computing `T_{p,k}` in characteristic `p`::
sage: p = 199
sage: fp = delta_qexp(prec=p^2+1, K=GF(p))
sage: tfp = hecke_operator_on_qexp(fp, p, 12)
sage: tfp == fp[p] * fp
True
sage: tf = hecke_operator_on_qexp(delta_qexp(prec=p^2+1), p, 12).change_ring(GF(p))
sage: tfp == tf
True
"""
if eps is None:
# Need to have base_ring=ZZ to work over finite fields, since
# ZZ can coerce to GF(p), but QQ can't.
eps = DirichletGroup(1, base_ring=ZZ)[0]
if check:
if not (is_PowerSeries(f) or is_ModularFormElement(f)):
raise TypeError("f (=%s) must be a power series or modular form" %
f)
if not is_DirichletCharacter(eps):
raise TypeError("eps (=%s) must be a Dirichlet character" % eps)
k = Integer(k)
n = Integer(n)
v = []
if prec is None:
if is_ModularFormElement(f):
# always want at least three coefficients, but not too many, unless
# requested
pr = max(f.prec(), f.parent().prec(), (n + 1) * 3)
pr = min(pr, 100 * (n + 1))
prec = pr // n + 1
else:
prec = (f.prec() / ZZ(n)).ceil()
if prec == Infinity: prec = f.parent().default_prec() // n + 1
if f.prec() < prec:
f._compute_q_expansion(prec)
p = Integer(f.base_ring().characteristic())
if k != 1 and p.is_prime() and n.is_power_of(p):
# if computing T_{p^a} in characteristic p, use the simpler (and faster)
# formula
v = [f[m * n] for m in range(prec)]
else:
l = k - 1
for m in range(prec):
am = sum([eps(d) * d**l * f[m*n//(d*d)] for \
d in divisors(gcd(n, m)) if (m*n) % (d*d) == 0])
v.append(am)
if _return_list:
return v
if is_ModularFormElement(f):
R = f.parent()._q_expansion_ring()
else:
R = f.parent()
return R(v, prec)
def hecke_operator_on_basis(B, n, k, eps=None, already_echelonized=False):
r"""
Given a basis `B` of `q`-expansions for a space of modular forms
with character `\varepsilon` to precision at least `\#B\cdot n+1`,
this function computes the matrix of `T_n` relative to `B`.
.. note::
If the elements of B are not known to sufficient precision,
this function will report that the vectors are linearly
dependent (since they are to the specified precision).
INPUT:
- ``B`` - list of q-expansions
- ``n`` - an integer >= 1
- ``k`` - an integer
- ``eps`` - Dirichlet character
- ``already_echelonized`` -- bool (default: False); if True, use that the
basis is already in Echelon form, which saves a lot of time.
EXAMPLES::
sage: sage.modular.modform.constructor.ModularForms_clear_cache()
sage: ModularForms(1,12).q_expansion_basis()
[
q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 + O(q^6),
1 + 65520/691*q + 134250480/691*q^2 + 11606736960/691*q^3 + 274945048560/691*q^4 + 3199218815520/691*q^5 + O(q^6)
]
sage: hecke_operator_on_basis(ModularForms(1,12).q_expansion_basis(), 3, 12)
Traceback (most recent call last):
...
ValueError: The given basis vectors must be linearly independent.
sage: hecke_operator_on_basis(ModularForms(1,12).q_expansion_basis(30), 3, 12)
[ 252 0]
[ 0 177148]
TESTS:
This shows that the problem with finite fields reported at trac #8281 is solved::
sage: bas_mod5 = [f.change_ring(GF(5)) for f in victor_miller_basis(12, 20)]
sage: hecke_operator_on_basis(bas_mod5, 2, 12)
[4 0]
[0 1]
This shows that empty input is handled sensibly (trac #12202)::
sage: x = hecke_operator_on_basis([], 3, 12); x
[]
sage: x.parent()
Full MatrixSpace of 0 by 0 dense matrices over Cyclotomic Field of order 1 and degree 1
sage: y = hecke_operator_on_basis([], 3, 12, eps=DirichletGroup(13).0^2); y
[]
sage: y.parent()
Full MatrixSpace of 0 by 0 dense matrices over Cyclotomic Field of order 12 and degree 4
"""
if not isinstance(B, (list, tuple)):
raise TypeError("B (=%s) must be a list or tuple" % B)
if len(B) == 0:
if eps is None:
R = CyclotomicField(1)
else:
R = eps.base_ring()
return MatrixSpace(R, 0)(0)
f = B[0]
R = f.base_ring()
if eps is None:
eps = DirichletGroup(1, R)[0]
all_powerseries = True
for x in B:
if not is_PowerSeries(x):
all_powerseries = False
if not all_powerseries:
raise TypeError("each element of B must be a power series")
n = Integer(n)
k = Integer(k)
prec = (f.prec() - 1) // n
A = R**prec
V = A.span_of_basis([g.padded_list(prec) for g in B],
already_echelonized=already_echelonized)
return _hecke_operator_on_basis(B, V, n, k, eps)