Modular Forms with Character¶
EXAMPLES:
sage: eps = DirichletGroup(13).0
sage: M = ModularForms(eps^2, 2); M
Modular Forms space of dimension 3, character [zeta6] and weight 2 over Cyclotomic Field of order 6 and degree 2
sage: S = M.cuspidal_submodule(); S
Cuspidal subspace of dimension 1 of Modular Forms space of dimension 3, character [zeta6] and weight 2 over Cyclotomic Field of order 6 and degree 2
sage: S.modular_symbols()
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
We create a spaces associated to Dirichlet characters of modulus 225:
sage: e = DirichletGroup(225).0
sage: e.order()
6
sage: e.base_ring()
Cyclotomic Field of order 60 and degree 16
sage: M = ModularForms(e,3)
Notice that the base ring is “minimized”:
sage: M
Modular Forms space of dimension 66, character [zeta6, 1] and weight 3
over Cyclotomic Field of order 6 and degree 2
If we don’t want the base ring to change, we can explicitly specify it:
sage: ModularForms(e, 3, e.base_ring())
Modular Forms space of dimension 66, character [zeta6, 1] and weight 3
over Cyclotomic Field of order 60 and degree 16
Next we create a space associated to a Dirichlet character of order 20:
sage: e = DirichletGroup(225).1
sage: e.order()
20
sage: e.base_ring()
Cyclotomic Field of order 60 and degree 16
sage: M = ModularForms(e,17); M
Modular Forms space of dimension 484, character [1, zeta20] and
weight 17 over Cyclotomic Field of order 20 and degree 8
We compute the Eisenstein subspace, which is fast even though the dimension of the space is large (since an explicit basis of \(q\)-expansions has not been computed yet).
sage: M.eisenstein_submodule()
Eisenstein subspace of dimension 8 of Modular Forms space of
dimension 484, character [1, zeta20] and weight 17 over Cyclotomic Field of order 20 and degree 8
sage: M.cuspidal_submodule()
Cuspidal subspace of dimension 476 of Modular Forms space of dimension 484, character [1, zeta20] and weight 17 over Cyclotomic Field of order 20 and degree 8
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class
sage.modular.modform.ambient_eps.ModularFormsAmbient_eps(character, weight=2, base_ring=None, eis_only=False)¶ Bases:
sage.modular.modform.ambient.ModularFormsAmbientA space of modular forms with character.
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change_ring(base_ring)¶ Return space with same defining parameters as this ambient space of modular symbols, but defined over a different base ring.
EXAMPLES:
sage: m = ModularForms(DirichletGroup(13).0^2,2); m Modular Forms space of dimension 3, character [zeta6] and weight 2 over Cyclotomic Field of order 6 and degree 2 sage: m.change_ring(CyclotomicField(12)) Modular Forms space of dimension 3, character [zeta6] and weight 2 over Cyclotomic Field of order 12 and degree 4
It must be possible to change the ring of the underlying Dirichlet character:
sage: m.change_ring(QQ) Traceback (most recent call last): ... TypeError: Unable to coerce zeta6 to a rational
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cuspidal_submodule()¶ Return the cuspidal submodule of this ambient space of modular forms.
EXAMPLES:
sage: eps = DirichletGroup(4).0 sage: M = ModularForms(eps, 5); M Modular Forms space of dimension 3, character [-1] and weight 5 over Rational Field sage: M.cuspidal_submodule() Cuspidal subspace of dimension 1 of Modular Forms space of dimension 3, character [-1] and weight 5 over Rational Field
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eisenstein_submodule()¶ Return the submodule of this ambient module with character that is spanned by Eisenstein series. This is the Hecke stable complement of the cuspidal submodule.
EXAMPLES:
sage: m = ModularForms(DirichletGroup(13).0^2,2); m Modular Forms space of dimension 3, character [zeta6] and weight 2 over Cyclotomic Field of order 6 and degree 2 sage: m.eisenstein_submodule() Eisenstein subspace of dimension 2 of Modular Forms space of dimension 3, character [zeta6] and weight 2 over Cyclotomic Field of order 6 and degree 2
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hecke_module_of_level(N)¶ Return the Hecke module of level N corresponding to self, which is the domain or codomain of a degeneracy map from self. Here N must be either a divisor or a multiple of the level of self, and a multiple of the conductor of the character of self.
EXAMPLES:
sage: M = ModularForms(DirichletGroup(15).0, 3); M.character().conductor() 3 sage: M.hecke_module_of_level(3) Modular Forms space of dimension 2, character [-1] and weight 3 over Rational Field sage: M.hecke_module_of_level(5) Traceback (most recent call last): ... ValueError: conductor(=3) must divide M(=5) sage: M.hecke_module_of_level(30) Modular Forms space of dimension 16, character [-1, 1] and weight 3 over Rational Field
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modular_symbols(sign=0)¶ Return corresponding space of modular symbols with given sign.
EXAMPLES:
sage: eps = DirichletGroup(13).0 sage: M = ModularForms(eps^2, 2) sage: M.modular_symbols() 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: M.modular_symbols(1) 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: M.modular_symbols(-1) Modular Symbols space of dimension 1 and level 13, weight 2, character [zeta6], sign -1, over Cyclotomic Field of order 6 and degree 2 sage: M.modular_symbols(2) Traceback (most recent call last): ... ValueError: sign must be -1, 0, or 1
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