# Ambient Spaces of Modular Forms¶

EXAMPLES:

We compute a basis for the ambient space $$M_2(\Gamma_1(25),\chi)$$, where $$\chi$$ is quadratic.

sage: chi = DirichletGroup(25,QQ).0; chi
Dirichlet character modulo 25 of conductor 5 mapping 2 |--> -1
sage: n = ModularForms(chi,2); n
Modular Forms space of dimension 6, character [-1] and weight 2 over Rational Field
sage: type(n)
<class 'sage.modular.modform.ambient_eps.ModularFormsAmbient_eps_with_category'>


Compute a basis:

sage: n.basis()
[
1 + O(q^6),
q + O(q^6),
q^2 + O(q^6),
q^3 + O(q^6),
q^4 + O(q^6),
q^5 + O(q^6)
]


Compute the same basis but to higher precision:

sage: n.set_precision(20)
sage: n.basis()
[
1 + 10*q^10 + 20*q^15 + O(q^20),
q + 5*q^6 + q^9 + 12*q^11 - 3*q^14 + 17*q^16 + 8*q^19 + O(q^20),
q^2 + 4*q^7 - q^8 + 8*q^12 + 2*q^13 + 10*q^17 - 5*q^18 + O(q^20),
q^3 + q^7 + 3*q^8 - q^12 + 5*q^13 + 3*q^17 + 6*q^18 + O(q^20),
q^4 - q^6 + 2*q^9 + 3*q^14 - 2*q^16 + 4*q^19 + O(q^20),
q^5 + q^10 + 2*q^15 + O(q^20)
]


TESTS:

sage: m = ModularForms(Gamma1(20),2,GF(7))
True

sage: m = ModularForms(GammaH(11,[4]), 2); m
Modular Forms space of dimension 2 for Congruence Subgroup Gamma_H(11) with H generated by [4] of weight 2 over Rational Field
sage: type(m)
<class 'sage.modular.modform.ambient_g1.ModularFormsAmbient_gH_Q_with_category'>
True

class sage.modular.modform.ambient.ModularFormsAmbient(group, weight, base_ring, character=None)

An ambient space of modular forms.

ambient_space()

Return the ambient space that contains this ambient space. This is, of course, just this space again.

EXAMPLES:

sage: m = ModularForms(Gamma0(3),30)
sage: m.ambient_space() is m
True

change_ring(base_ring)

Change the base ring of this space of modular forms.

INPUT:

• R - ring

EXAMPLES:

sage: M = ModularForms(Gamma0(37),2)
sage: M.basis()
[
q + q^3 - 2*q^4 + O(q^6),
q^2 + 2*q^3 - 2*q^4 + q^5 + O(q^6),
1 + 2/3*q + 2*q^2 + 8/3*q^3 + 14/3*q^4 + 4*q^5 + O(q^6)
]


The basis after changing the base ring is the reduction modulo $$3$$ of an integral basis.

sage: M3 = M.change_ring(GF(3))
sage: M3.basis()
[
q + q^3 + q^4 + O(q^6),
q^2 + 2*q^3 + q^4 + q^5 + O(q^6),
1 + q^3 + q^4 + 2*q^5 + O(q^6)
]

cuspidal_submodule()

Return the cuspidal submodule of this ambient module.

EXAMPLES:

sage: ModularForms(Gamma1(13)).cuspidal_submodule()
Cuspidal subspace of dimension 2 of Modular Forms space of dimension 13 for
Congruence Subgroup Gamma1(13) of weight 2 over Rational Field

dimension()

Return the dimension of this ambient space of modular forms, computed using a dimension formula (so it should be reasonably fast).

EXAMPLES:

sage: m = ModularForms(Gamma1(20),20)
sage: m.dimension()
238

eisenstein_params()

Return parameters that define all Eisenstein series in self.

OUTPUT: an immutable Sequence

EXAMPLES:

sage: m = ModularForms(Gamma0(22), 2)
sage: v = m.eisenstein_params(); v
[(Dirichlet character modulo 22 of conductor 1 mapping 13 |--> 1, Dirichlet character modulo 22 of conductor 1 mapping 13 |--> 1, 2), (Dirichlet character modulo 22 of conductor 1 mapping 13 |--> 1, Dirichlet character modulo 22 of conductor 1 mapping 13 |--> 1, 11), (Dirichlet character modulo 22 of conductor 1 mapping 13 |--> 1, Dirichlet character modulo 22 of conductor 1 mapping 13 |--> 1, 22)]
sage: type(v)
<class 'sage.structure.sequence.Sequence_generic'>

eisenstein_series()

Return all Eisenstein series associated to this space.

sage: ModularForms(27,2).eisenstein_series()
[
q^3 + O(q^6),
q - 3*q^2 + 7*q^4 - 6*q^5 + O(q^6),
1/12 + q + 3*q^2 + q^3 + 7*q^4 + 6*q^5 + O(q^6),
1/3 + q + 3*q^2 + 4*q^3 + 7*q^4 + 6*q^5 + O(q^6),
13/12 + q + 3*q^2 + 4*q^3 + 7*q^4 + 6*q^5 + O(q^6)
]

sage: ModularForms(Gamma1(5),3).eisenstein_series()
[
-1/5*zeta4 - 2/5 + q + (4*zeta4 + 1)*q^2 + (-9*zeta4 + 1)*q^3 + (4*zeta4 - 15)*q^4 + q^5 + O(q^6),
q + (zeta4 + 4)*q^2 + (-zeta4 + 9)*q^3 + (4*zeta4 + 15)*q^4 + 25*q^5 + O(q^6),
1/5*zeta4 - 2/5 + q + (-4*zeta4 + 1)*q^2 + (9*zeta4 + 1)*q^3 + (-4*zeta4 - 15)*q^4 + q^5 + O(q^6),
q + (-zeta4 + 4)*q^2 + (zeta4 + 9)*q^3 + (-4*zeta4 + 15)*q^4 + 25*q^5 + O(q^6)
]

sage: eps = DirichletGroup(13).0^2
sage: ModularForms(eps,2).eisenstein_series()
[
-7/13*zeta6 - 11/13 + q + (2*zeta6 + 1)*q^2 + (-3*zeta6 + 1)*q^3 + (6*zeta6 - 3)*q^4 - 4*q^5 + O(q^6),
q + (zeta6 + 2)*q^2 + (-zeta6 + 3)*q^3 + (3*zeta6 + 3)*q^4 + 4*q^5 + O(q^6)
]

eisenstein_submodule()

Return the Eisenstein submodule of this ambient module.

EXAMPLES:

sage: m = ModularForms(Gamma1(13),2); m
Modular Forms space of dimension 13 for Congruence Subgroup Gamma1(13) of weight 2 over Rational Field
sage: m.eisenstein_submodule()
Eisenstein subspace of dimension 11 of Modular Forms space of dimension 13 for Congruence Subgroup Gamma1(13) of weight 2 over Rational Field

free_module()

Return the free module underlying this space of modular forms.

EXAMPLES:

sage: ModularForms(37).free_module()
Vector space of dimension 3 over Rational Field

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.

EXAMPLES:

sage: ModularForms(25, 6).hecke_module_of_level(5)
Modular Forms space of dimension 3 for Congruence Subgroup Gamma0(5) of weight 6 over Rational Field
sage: ModularForms(Gamma1(4), 3).hecke_module_of_level(8)
Modular Forms space of dimension 7 for Congruence Subgroup Gamma1(8) of weight 3 over Rational Field
sage: ModularForms(Gamma1(4), 3).hecke_module_of_level(9)
Traceback (most recent call last):
...
ValueError: N (=9) must be a divisor or a multiple of the level of self (=4)

is_ambient()

Return True if this an ambient space of modular forms.

This is an ambient space, so this function always returns True.

EXAMPLES:

sage: ModularForms(11).is_ambient()
True
sage: CuspForms(11).is_ambient()
False

modular_symbols(sign=0)

Return the corresponding space of modular symbols with the given sign.

EXAMPLES:

sage: S = ModularForms(11,2)
sage: S.modular_symbols()
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 space of dimension 2 for Gamma_0(11) of weight 2 with sign 1 over Rational Field
sage: S.modular_symbols(sign=-1)
Modular Symbols space of dimension 1 for Gamma_0(11) of weight 2 with sign -1 over Rational Field

sage: ModularForms(1,12).modular_symbols()
Modular Symbols space of dimension 3 for Gamma_0(1) of weight 12 with sign 0 over Rational Field

module()

Return the underlying free module corresponding to this space of modular forms.

If the dimension of self can be computed reasonably quickly, then this function returns a free module (viewed as a tuple space) of the same dimension as self over the same base ring. Otherwise, the dimension of self.module() may be smaller. For example, in the case of weight 1 forms, in some cases the dimension can’t easily be computed so self.module() is of smaller dimension.

EXAMPLES:

sage: m = ModularForms(Gamma1(13),10)
sage: m.free_module()
Vector space of dimension 69 over Rational Field
sage: ModularForms(Gamma1(13),4, GF(49,'b')).free_module()
Vector space of dimension 27 over Finite Field in b of size 7^2


Note that in the following example the dimension can’t be (quickly) computed, so M.module() returns a space of different dimension than M:

sage: M = ModularForms(Gamma1(57), 1); M
Modular Forms space of dimension (unknown) for Congruence ...
sage: M.module()
Vector space of dimension 36 over Rational Field
sage: M.basis()
<repr(<sage.structure.sequence.Sequence_generic at 0x...>) failed: NotImplementedError: Computation of dimensions of weight 1 cusp forms spaces not implemented in general>

new_submodule(p=None)

Return the new or $$p$$-new submodule of this ambient module.

INPUT:

• p - (default: None), if specified return only the $$p$$-new submodule.

EXAMPLES:

sage: m = ModularForms(Gamma0(33),2); m
Modular Forms space of dimension 6 for Congruence Subgroup Gamma0(33) of weight 2 over Rational Field
sage: m.new_submodule()
Modular Forms subspace of dimension 1 of Modular Forms space of dimension 6 for Congruence Subgroup Gamma0(33) of weight 2 over Rational Field


Another example:

sage: M = ModularForms(17,4)
sage: N = M.new_subspace(); N
Modular Forms subspace of dimension 4 of Modular Forms space of dimension 6 for Congruence Subgroup Gamma0(17) of weight 4 over Rational Field
sage: N.basis()
[
q + 2*q^5 + O(q^6),
q^2 - 3/2*q^5 + O(q^6),
q^3 + O(q^6),
q^4 - 1/2*q^5 + O(q^6)
]

sage: ModularForms(12,4).new_submodule()
Modular Forms subspace of dimension 1 of Modular Forms space of dimension 9 for Congruence Subgroup Gamma0(12) of weight 4 over Rational Field


Unfortunately (TODO) - $$p$$-new submodules aren’t yet implemented:

sage: m.new_submodule(3)            # not implemented
Traceback (most recent call last):
...
NotImplementedError
sage: m.new_submodule(11)           # not implemented
Traceback (most recent call last):
...
NotImplementedError

prec(new_prec=None)

Set or get default initial precision for printing modular forms.

INPUT:

• new_prec - positive integer (default: None)

OUTPUT: if new_prec is None, returns the current precision.

EXAMPLES:

sage: M = ModularForms(1,12, prec=3)
sage: M.prec()
3

sage: M.basis()
[
q - 24*q^2 + O(q^3),
1 + 65520/691*q + 134250480/691*q^2 + O(q^3)
]

sage: M.prec(5)
5
sage: M.basis()
[
q - 24*q^2 + 252*q^3 - 1472*q^4 + O(q^5),
1 + 65520/691*q + 134250480/691*q^2 + 11606736960/691*q^3 + 274945048560/691*q^4 + O(q^5)
]

rank()

This is a synonym for self.dimension().

EXAMPLES:

sage: m = ModularForms(Gamma0(20),4)
sage: m.rank()
12
sage: m.dimension()
12

set_precision(n)

Set the default precision for displaying elements of this space.

EXAMPLES:

sage: m = ModularForms(Gamma1(5),2)
sage: m.set_precision(10)
sage: m.basis()
[
1 + 60*q^3 - 120*q^4 + 240*q^5 - 300*q^6 + 300*q^7 - 180*q^9 + O(q^10),
q + 6*q^3 - 9*q^4 + 27*q^5 - 28*q^6 + 30*q^7 - 11*q^9 + O(q^10),
q^2 - 4*q^3 + 12*q^4 - 22*q^5 + 30*q^6 - 24*q^7 + 5*q^8 + 18*q^9 + O(q^10)
]
sage: m.set_precision(5)
sage: m.basis()
[
1 + 60*q^3 - 120*q^4 + O(q^5),
q + 6*q^3 - 9*q^4 + O(q^5),
q^2 - 4*q^3 + 12*q^4 + O(q^5)
]


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