# 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


TESTS:

sage: m = ModularForms(DirichletGroup(20).1,5)
True
sage: type(m)
<class 'sage.modular.modform.ambient_eps.ModularFormsAmbient_eps_with_category'>

class sage.modular.modform.ambient_eps.ModularFormsAmbient_eps(character, weight=2, base_ring=None)

A space of modular forms with character.

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):
...
ValueError: cannot coerce element of order 6 into self

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

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

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

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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