Module: sage.modular.modform.cuspidal_submodule
The Cuspidal Subspace
sage: S = CuspForms(SL2Z,12); S Cuspidal subspace of dimension 1 of Modular Forms space of dimension 2 for Congruence Subgroup Gamma0(1) of weight 12 over Rational Field sage: S.basis() [ q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 + O(q^6) ]
sage: S = CuspForms(Gamma0(33),2); S Cuspidal subspace of dimension 3 of Modular Forms space of dimension 6 for Congruence Subgroup Gamma0(33) of weight 2 over Rational Field sage: S.basis() [ q - q^5 + O(q^6), q^2 - q^4 - q^5 + O(q^6), q^3 + O(q^6) ]
sage: S = CuspForms(Gamma1(3),6); S Cuspidal subspace of dimension 1 of Modular Forms space of dimension 3 for Congruence Subgroup Gamma1(3) of weight 6 over Rational Field sage: S.basis() [ q - 6*q^2 + 9*q^3 + 4*q^4 + 6*q^5 + O(q^6) ]
Class: CuspidalSubmodule
self, ambient_space) |
The cuspidal submodule of an ambient space of modular forms.
sage: S = CuspForms(SL2Z,12); S Cuspidal subspace of dimension 1 of Modular Forms space of dimension 2 for Congruence Subgroup Gamma0(1) of weight 12 over Rational Field sage: S.basis() [ q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 + O(q^6) ]
sage: S = CuspForms(Gamma0(33),2); S Cuspidal subspace of dimension 3 of Modular Forms space of dimension 6 for Congruence Subgroup Gamma0(33) of weight 2 over Rational Field sage: S.basis() [ q - q^5 + O(q^6), q^2 - q^4 - q^5 + O(q^6), q^3 + O(q^6) ]
sage: S = CuspForms(Gamma1(3),6); S Cuspidal subspace of dimension 1 of Modular Forms space of dimension 3 for Congruence Subgroup Gamma1(3) of weight 6 over Rational Field sage: S.basis() [ q - 6*q^2 + 9*q^3 + 4*q^4 + 6*q^5 + O(q^6) ] sage: S == loads(dumps(S)) True
Functions: modular_symbols
self, [sign=0]) |
Return the corresponding space of modular symbols with the given sign.
sage: S = ModularForms(11,2).cuspidal_submodule() sage: S.modular_symbols() Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 3 for Gamma_0(11) of weight 2 with sign 0 over Rational Field
sage: S.modular_symbols(sign=-1) Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 1 for Gamma_0(11) of weight 2 with sign -1 over Rational Field
sage: M = S.modular_symbols(sign=1); M Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 2 for Gamma_0(11) of weight 2 with sign 1 over Rational Field sage: M.sign() 1
sage: S = ModularForms(1,12).cuspidal_submodule() sage: S.modular_symbols() Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 3 for Gamma_0(1) of weight 12 with sign 0 over Rational Field
sage: eps = DirichletGroup(13).0 sage: S = CuspForms(eps^2, 2)
sage: S.modular_symbols(sign=0) Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 4 and level 13, weight 2, character [zeta6], sign 0, over Cyclotomic Field of order 6 and degree 2
sage: S.modular_symbols(sign=1) Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 3 and level 13, weight 2, character [zeta6], sign 1, over Cyclotomic Field of order 6 and degree 2
sage: S.modular_symbols(sign=-1) Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 1 and level 13, weight 2, character [zeta6], sign -1, over Cyclotomic Field of order 6 and degree 2
Special Functions: __init__,
_repr_
self) |
Return the string representation of self.
sage: S = CuspForms(Gamma1(3),6); S._repr_() 'Cuspidal subspace of dimension 1 of Modular Forms space of dimension 3 for Congruence Subgroup Gamma1(3) of weight 6 over Rational Field'
Class: CuspidalSubmodule_eps
sage: S = CuspForms(DirichletGroup(5).0,5); S Cuspidal subspace of dimension 1 of Modular Forms space of dimension 3, character [zeta4] and weight 5 over Cyclotomic Field of order 4 and degree 2
sage: S.basis() [ q + (-zeta4 - 1)*q^2 + (6*zeta4 - 6)*q^3 - 14*zeta4*q^4 + (15*zeta4 + 20)*q^5 + O(q^6) ] sage: f = S.0 sage: f.qexp() q + (-zeta4 - 1)*q^2 + (6*zeta4 - 6)*q^3 - 14*zeta4*q^4 + (15*zeta4 + 20)*q^5 + O(q^6) sage: f.qexp(7) q + (-zeta4 - 1)*q^2 + (6*zeta4 - 6)*q^3 - 14*zeta4*q^4 + (15*zeta4 + 20)*q^5 + 12*q^6 + O(q^7) sage: f.qexp(3) q + (-zeta4 - 1)*q^2 + O(q^3) sage: f.qexp(2) q + O(q^2) sage: f.qexp(1) O(q^1)
Class: CuspidalSubmodule_g0_Q
Class: CuspidalSubmodule_g1_Q
Class: CuspidalSubmodule_level1_Q
Special Functions: _compute_q_expansion_basis
self, [prec=None]) |
Compute q-expansions of a basis for self.
sage: sage.modular.modform.cuspidal_submodule.CuspidalSubmodule_level1_Q(ModularForms(1,12))._compute_q_expansion_basis() [ q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 + O(q^6) ]
Class: CuspidalSubmodule_modsym_qexp
Special Functions: _compute_q_expansion_basis
self, [prec=None]) |
Compute q-expansions of a basis for self (via modular symbols).
sage: sage.modular.modform.cuspidal_submodule.CuspidalSubmodule_modsym_qexp(ModularForms(11,2))._compute_q_expansion_basis() [ q - 2*q^2 - q^3 + 2*q^4 + q^5 + O(q^6) ]