42.9 The Cuspidal Subspace

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

class CuspidalSubmodule
The cuspidal submodule of an ambient space of modular forms.
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

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_

_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

class CuspidalSubmodule_eps
Space of cusp forms with given Dirichlet character.

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_g0_Q
Space of cusp forms for Gamma0(N).

Class: CuspidalSubmodule_g1_Q

class CuspidalSubmodule_g1_Q
Space of cusp forms for Gamma1(N).

Class: CuspidalSubmodule_level1_Q

class CuspidalSubmodule_level1_Q
Space of cusp forms of level 1 over Q.

Special Functions: _compute_q_expansion_basis

_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

class CuspidalSubmodule_modsym_qexp
Cuspidal submodule with q-expansions calculated via modular symbols.

Special Functions: _compute_q_expansion_basis

_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)
]

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