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r""" 

Series constructor for modular forms for Hecke triangle groups 

 

AUTHORS: 

 

- Based on the thesis of John Garrett Leo (2008) 

- Jonas Jermann (2013): initial version 

 

.. NOTE:: 

 

``J_inv_ZZ`` is the main function used to determine all Fourier expansions. 

""" 

from __future__ import absolute_import 

 

#***************************************************************************** 

# Copyright (C) 2013-2014 Jonas Jermann <jjermann2@gmail.com> 

# 

# Distributed under the terms of the GNU General Public License (GPL) 

# as published by the Free Software Foundation; either version 2 of 

# the License, or (at your option) any later version. 

# http://www.gnu.org/licenses/ 

#***************************************************************************** 

 

from sage.rings.all import ZZ, QQ, infinity, PolynomialRing, LaurentSeries, PowerSeriesRing, FractionField 

from sage.rings.big_oh import O 

from sage.functions.all import exp 

from sage.arith.all import bernoulli, sigma, rising_factorial 

 

from sage.structure.sage_object import SageObject 

from sage.structure.unique_representation import UniqueRepresentation 

from sage.misc.cachefunc import cached_method 

 

from .hecke_triangle_groups import HeckeTriangleGroup 

 

 

 

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())