Coverage for local/lib/python2.7/site-packages/sage/modular/modform/ambient_g1.py : 59%

r""" 

Modular Forms for `\Gamma_1(N)` and `\Gamma_H(N)` over `\QQ` 

EXAMPLES:: 

sage: M = ModularForms(Gamma1(13),2); M 

Modular Forms space of dimension 13 for Congruence Subgroup Gamma1(13) of weight 2 over Rational Field 

sage: S = M.cuspidal_submodule(); S 

Cuspidal subspace of dimension 2 of Modular Forms space of dimension 13 for Congruence Subgroup Gamma1(13) of weight 2 over Rational Field 

sage: S.basis() 

[ 

q - 4*q^3 - q^4 + 3*q^5 + O(q^6), 

q^2 - 2*q^3 - q^4 + 2*q^5 + O(q^6) 

] 

sage: M = ModularForms(GammaH(11, [3])); M 

Modular Forms space of dimension 2 for Congruence Subgroup Gamma_H(11) with H generated by [3] of weight 2 over Rational Field 

sage: M.q_expansion_basis(8) 

[ 

q - 2*q^2 - q^3 + 2*q^4 + q^5 + 2*q^6 - 2*q^7 + O(q^8), 

1 + 12/5*q + 36/5*q^2 + 48/5*q^3 + 84/5*q^4 + 72/5*q^5 + 144/5*q^6 + 96/5*q^7 + O(q^8) 

] 

TESTS:: 

sage: m = ModularForms(Gamma1(20),2) 

sage: loads(dumps(m)) == m 

True 

sage: m = ModularForms(GammaH(15, [4]), 2) 

sage: loads(dumps(m)) == m 

True 

We check that :trac:`10453` is fixed:: 

sage: CuspForms(Gamma1(11), 2).old_submodule() 

Modular Forms subspace of dimension 0 of Modular Forms space of dimension 10 for Congruence Subgroup Gamma1(11) of weight 2 over Rational Field 

sage: ModularForms(Gamma1(3), 12).old_submodule() 

Modular Forms subspace of dimension 4 of Modular Forms space of dimension 5 for Congruence Subgroup Gamma1(3) of weight 12 over Rational Field 

""" 

from __future__ import absolute_import 

######################################################################### 

# Copyright (C) 2006 William Stein <wstein@gmail.com> 

# 

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

# 

# http://www.gnu.org/licenses/ 

######################################################################### 

import sage.rings.all as rings 

import sage.modular.arithgroup.all as arithgroup 

from . import ambient 

from . import cuspidal_submodule 

from . import eisenstein_submodule 

class ModularFormsAmbient_gH_Q(ambient.ModularFormsAmbient): 

""" 

A space of modular forms for the group `\Gamma_H(N)` over the rational numbers. 

""" 

def __init__(self, group, weight, eis_only): 

r""" 

Create a space of modular forms for `\Gamma_H(N)` of integral weight over the 

rational numbers. 

EXAMPLES:: 

sage: m = ModularForms(GammaH(100, [41]),5); m 

Modular Forms space of dimension 270 for Congruence Subgroup Gamma_H(100) with H generated by [41] of weight 5 over Rational Field 

sage: type(m) 

<class 'sage.modular.modform.ambient_g1.ModularFormsAmbient_gH_Q_with_category'> 

""" 

ambient.ModularFormsAmbient.__init__(self, group, weight, rings.QQ, eis_only=eis_only) 

#################################################################### 

# Computation of Special Submodules 

#################################################################### 

def cuspidal_submodule(self): 

""" 

Return the cuspidal submodule of this modular forms space. 

EXAMPLES:: 

sage: m = ModularForms(GammaH(100, [29]),2); m 

Modular Forms space of dimension 48 for Congruence Subgroup Gamma_H(100) with H generated by [29] of weight 2 over Rational Field 

sage: m.cuspidal_submodule() 

Cuspidal subspace of dimension 13 of Modular Forms space of dimension 48 for Congruence Subgroup Gamma_H(100) with H generated by [29] of weight 2 over Rational Field 

""" 

try: 

return self.__cuspidal_submodule 

except AttributeError: 

if self.level() == 1: 

self.__cuspidal_submodule = cuspidal_submodule.CuspidalSubmodule_level1_Q(self) 

else: 

self.__cuspidal_submodule = cuspidal_submodule.CuspidalSubmodule_gH_Q(self) 

return self.__cuspidal_submodule 

def eisenstein_submodule(self): 

""" 

Return the Eisenstein submodule of this modular forms space. 

EXAMPLES:: 

sage: E = ModularForms(GammaH(100, [29]),3).eisenstein_submodule(); E 

Eisenstein subspace of dimension 24 of Modular Forms space of dimension 72 for Congruence Subgroup Gamma_H(100) with H generated by [29] of weight 3 over Rational Field 

sage: type(E) 

<class 'sage.modular.modform.eisenstein_submodule.EisensteinSubmodule_gH_Q_with_category'> 

""" 

try: 

return self.__eisenstein_submodule 

except AttributeError: 

self.__eisenstein_submodule = eisenstein_submodule.EisensteinSubmodule_gH_Q(self) 

return self.__eisenstein_submodule 

def _compute_diamond_matrix(self, d): 

r""" 

Compute the matrix of the diamond operator <d> on this space. 

EXAMPLES:: 

sage: ModularForms(GammaH(9, [4]), 7)._compute_diamond_matrix(2) 

[-1 0 0 0 0 0 0 0] 

[ 0 -1 0 0 0 0 0 0] 

[ 0 0 -1 0 0 0 0 0] 

[ 0 0 0 -1 0 0 0 0] 

[ 0 0 0 0 -1 0 0 0] 

[ 0 0 0 0 0 -1 0 0] 

[ 0 0 0 0 0 0 -1 0] 

[ 0 0 0 0 0 0 0 -1] 

""" 

return self.cuspidal_submodule().diamond_bracket_matrix(d).block_sum(self.eisenstein_submodule().diamond_bracket_matrix(d)) 

def _compute_hecke_matrix(self, n): 

r""" 

Compute the matrix of the Hecke operator T_n acting on this space. 

EXAMPLES:: 

sage: ModularForms(Gamma1(7), 4).hecke_matrix(3) # indirect doctest 

[ 0 -42 133 0 0 0 0 0 0] 

[ 0 -28 91 0 0 0 0 0 0] 

[ 1 -8 19 0 0 0 0 0 0] 

[ 0 0 0 28 0 0 0 0 0] 

[ 0 0 0 -10152/259 0 5222/37 -13230/37 -22295/37 92504/37] 

[ 0 0 0 -6087/259 0 312067/4329 1370420/4329 252805/333 3441466/4329] 

[ 0 0 0 -729/259 1 485/37 3402/37 5733/37 7973/37] 

[ 0 0 0 729/259 0 -189/37 -1404/37 -2366/37 -3348/37] 

[ 0 0 0 255/259 0 -18280/4329 -51947/4329 -10192/333 -190855/4329] 

""" 

return self.cuspidal_submodule().hecke_matrix(n).block_sum(self.eisenstein_submodule().hecke_matrix(n)) 

class ModularFormsAmbient_g1_Q(ModularFormsAmbient_gH_Q): 

""" 

A space of modular forms for the group `\Gamma_1(N)` over the rational numbers. 

""" 

def __init__(self, level, weight, eis_only): 

r""" 

Create a space of modular forms for `\Gamma_1(N)` of integral weight over the 

rational numbers. 

EXAMPLES:: 

sage: m = ModularForms(Gamma1(100),5); m 

Modular Forms space of dimension 1270 for Congruence Subgroup Gamma1(100) of weight 5 over Rational Field 

sage: type(m) 

<class 'sage.modular.modform.ambient_g1.ModularFormsAmbient_g1_Q_with_category'> 

""" 

ambient.ModularFormsAmbient.__init__(self, arithgroup.Gamma1(level), weight, rings.QQ, eis_only=eis_only) 

#################################################################### 

# Computation of Special Submodules 

#################################################################### 

def cuspidal_submodule(self): 

""" 

Return the cuspidal submodule of this modular forms space. 

EXAMPLES:: 

sage: m = ModularForms(Gamma1(17),2); m 

Modular Forms space of dimension 20 for Congruence Subgroup Gamma1(17) of weight 2 over Rational Field 

sage: m.cuspidal_submodule() 

Cuspidal subspace of dimension 5 of Modular Forms space of dimension 20 for Congruence Subgroup Gamma1(17) of weight 2 over Rational Field 

""" 

try: 

return self.__cuspidal_submodule 

except AttributeError: 

if self.level() == 1: 

self.__cuspidal_submodule = cuspidal_submodule.CuspidalSubmodule_level1_Q(self) 

else: 

self.__cuspidal_submodule = cuspidal_submodule.CuspidalSubmodule_g1_Q(self) 

return self.__cuspidal_submodule 

def eisenstein_submodule(self): 

""" 

Return the Eisenstein submodule of this modular forms space. 

EXAMPLES:: 

sage: ModularForms(Gamma1(13),2).eisenstein_submodule() 

Eisenstein subspace of dimension 11 of Modular Forms space of dimension 13 for Congruence Subgroup Gamma1(13) of weight 2 over Rational Field 

sage: ModularForms(Gamma1(13),10).eisenstein_submodule() 

Eisenstein subspace of dimension 12 of Modular Forms space of dimension 69 for Congruence Subgroup Gamma1(13) of weight 10 over Rational Field 

""" 

try: 

return self.__eisenstein_submodule 

except AttributeError: 

self.__eisenstein_submodule = eisenstein_submodule.EisensteinSubmodule_g1_Q(self) 

return self.__eisenstein_submodule