EXAMPLES:
sage: S = CuspForms(SL2Z,12); S
Cuspidal subspace of dimension 1 of Modular Forms space of dimension 2 for
Modular Group SL(2,Z) 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)
]
Bases: sage.modular.modform.submodule.ModularFormsSubmodule
Base class for cuspidal submodules of ambient spaces of modular forms.
Change the base ring of self to R, when this makes sense. This differs from base_extend() in that there may not be a canonical map from self to the new space, as in the first example below. If this space has a character then this may fail when the character cannot be defined over R, as in the second example.
EXAMPLES:
sage: chi = DirichletGroup(109, CyclotomicField(3)).0
sage: S9 = CuspForms(chi, 2, base_ring = CyclotomicField(9)); S9
Cuspidal subspace of dimension 8 of Modular Forms space of dimension 10, character [zeta3 + 1] and weight 2 over Cyclotomic Field of order 9 and degree 6
sage: S9.change_ring(CyclotomicField(3))
Cuspidal subspace of dimension 8 of Modular Forms space of dimension 10, character [zeta3 + 1] and weight 2 over Cyclotomic Field of order 3 and degree 2
sage: S9.change_ring(QQ)
...
ValueError: Space cannot be defined over Rational Field
Return True since spaces of cusp forms are cuspidal.
EXAMPLES:
sage: CuspForms(4,10).is_cuspidal()
True
Return the corresponding space of modular symbols with the given sign.
EXAMPLES:
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
Bases: sage.modular.modform.cuspidal_submodule.CuspidalSubmodule
Cuspidal submodule over a non-minimal base ring.
Bases: sage.modular.modform.cuspidal_submodule.CuspidalSubmodule_modsym_qexp
Space of cusp forms with given Dirichlet character.
EXAMPLES:
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)
Bases: sage.modular.modform.cuspidal_submodule.CuspidalSubmodule_modsym_qexp
Space of cusp forms for over .
Bases: sage.modular.modform.cuspidal_submodule.CuspidalSubmodule_modsym_qexp
Space of cusp forms for over .
Bases: sage.modular.modform.cuspidal_submodule.CuspidalSubmodule
Space of cusp forms of level 1 over .
Bases: sage.modular.modform.cuspidal_submodule.CuspidalSubmodule
Cuspidal submodule with q-expansions calculated via modular symbols.
Return the new subspace of this space of cusp forms. This is computed using modular symbols.
EXAMPLE:
sage: CuspForms(55).new_submodule()
Modular Forms subspace of dimension 3 of Modular Forms space of dimension 8 for Congruence Subgroup Gamma0(55) of weight 2 over Rational Field