# Modular Forms over a Non-minimal Base Ring¶

class sage.modular.modform.ambient_R.ModularFormsAmbient_R(M, base_ring)

Ambient space of modular forms over a ring other than QQ.

EXAMPLES:

sage: M = ModularForms(23,2,base_ring=GF(7)) ## indirect doctest
sage: M
Modular Forms space of dimension 3 for Congruence Subgroup Gamma0(23) of weight 2 over Finite Field of size 7
True

change_ring(R)

Return this modular forms space with the base ring changed to the ring R.

EXAMPLE:

sage: chi = DirichletGroup(109, CyclotomicField(3)).0
sage: M9 = ModularForms(chi, 2, base_ring = CyclotomicField(9))
sage: M9.change_ring(CyclotomicField(15))
Modular Forms space of dimension 10, character [zeta3 + 1] and weight 2 over Cyclotomic Field of order 15 and degree 8
sage: M9.change_ring(QQ)
Traceback (most recent call last):
...
ValueError: Space cannot be defined over Rational Field

cuspidal_submodule()

Return the cuspidal subspace of this space.

EXAMPLE:

sage: C = CuspForms(7, 4, base_ring=CyclotomicField(5)) # indirect doctest
sage: type(C)
<class 'sage.modular.modform.cuspidal_submodule.CuspidalSubmodule_R_with_category'>

modular_symbols(sign=0)

Return the space of modular symbols attached to this space, with the given sign (default 0).

TESTS:

sage: K.<i> = QuadraticField(-1)
sage: chi = DirichletGroup(5, base_ring = K).0
sage: L.<c> = K.extension(x^2 - 402*i)
sage: M = ModularForms(chi, 7, base_ring = L)
sage: symbs = M.modular_symbols()
sage: symbs.character() == chi
True
sage: symbs.base_ring() == L
True


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Modular Forms for $$\Gamma_1(N)$$ and $$\Gamma_H(N)$$ over $$\QQ$$

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Submodules of spaces of modular forms