def centralizer_group_card(self, q=None):
"""
Return the cardinality of the centralizer group of a matrix of type
``self`` in a field of order ``q``.
INPUT:
``q`` -- an integer or an indeterminate
EXAMPLES::
sage: PT = PrimarySimilarityClassType(1, [])
sage: PT.centralizer_group_card()
1
sage: PT = PrimarySimilarityClassType(2, [1, 1])
sage: PT.centralizer_group_card()
q^8 - q^6 - q^4 + q^2
"""
if q == None:
R = FractionField(ZZ["q"])
q = R.gen()
return self.statistic(centralizer_group_cardinality, q=q)
def centralizer_group_card(self, q=None):
"""
Return the cardinality of the centralizer group of a matrix of type
``self`` in a field of order ``q``.
INPUT:
``q`` -- an integer or an indeterminate
EXAMPLES::
sage: PT = PrimarySimilarityClassType(1, [])
sage: PT.centralizer_group_card()
1
sage: PT = PrimarySimilarityClassType(2, [1, 1])
sage: PT.centralizer_group_card()
q^8 - q^6 - q^4 + q^2
"""
if q == None:
R = FractionField(ZZ['q'])
q = R.gen()
return self.statistic(centralizer_group_cardinality, q=q)
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)
