Ambient Hecke modules¶

class sage.modular.hecke.ambient_module.AmbientHeckeModule(base_ring, rank, level, weight, category=None)[source]¶

Bases: HeckeModule_free_module

An ambient Hecke module, i.e. a Hecke module that is isomorphic as a module over its base ring \(R\) to the standard free module \(R^k\) for some \(k\). This is the base class for ambient spaces of modular forms and modular symbols, and for Brandt modules.

ambient_hecke_module()[source]¶

Return the ambient space that contains this ambient space.

This is, of course, just this space again.

EXAMPLES:

Sage

sage: M = ModularForms(11, 4); M.ambient_hecke_module() is M
True

Python

>>> from sage.all import *
>>> M = ModularForms(Integer(11), Integer(4)); M.ambient_hecke_module() is M
True

complement()[source]¶

Return the largest Hecke-stable complement of this space.

EXAMPLES:

Sage

sage: M = ModularSymbols(11,2,1); M
Modular Symbols space of dimension 2 for Gamma_0(11) of weight 2 with sign 1 over Rational Field
sage: M.complement()
Modular Symbols subspace of dimension 0 of Modular Symbols space of dimension 2 for Gamma_0(11) of weight 2 with sign 1 over Rational Field
sage: C = M.cuspidal_subspace(); C
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: C.complement()
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

Python

>>> from sage.all import *
>>> M = ModularSymbols(Integer(11),Integer(2),Integer(1)); M
Modular Symbols space of dimension 2 for Gamma_0(11) of weight 2 with sign 1 over Rational Field
>>> M.complement()
Modular Symbols subspace of dimension 0 of Modular Symbols space of dimension 2 for Gamma_0(11) of weight 2 with sign 1 over Rational Field
>>> C = M.cuspidal_subspace(); C
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
>>> C.complement()
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

decomposition_matrix()[source]¶

Return the matrix whose columns form a basis for the canonical sorted decomposition of self coming from the Hecke operators.

If the simple factors are \(D_0, \ldots, D_n\), then the first few columns are an echelonized basis for \(D_0\), the next an echelonized basis for \(D_1\), the next for \(D_2\), etc.

EXAMPLES:

Sage

sage: S = ModularSymbols(37, 2)
sage: S.decomposition_matrix()
[   1    0    0    0 -1/3]
[   0    1   -1    0  1/2]
[   0    0    0    1 -1/2]
[   0    1    1    1    0]
[   0    0    0    0    1]

Python

>>> from sage.all import *
>>> S = ModularSymbols(Integer(37), Integer(2))
>>> S.decomposition_matrix()
[   1    0    0    0 -1/3]
[   0    1   -1    0  1/2]
[   0    0    0    1 -1/2]
[   0    1    1    1    0]
[   0    0    0    0    1]

decomposition_matrix_inverse()[source]¶

Return the inverse of the decomposition matrix.

This is the inverse of the matrix returned by decomposition_matrix().

EXAMPLES:

Sage

sage: S = ModularSymbols(37, 2)
sage: t = S.decomposition_matrix_inverse(); t
[   1    0    0    0  1/3]
[   0  1/2 -1/2  1/2 -1/2]
[   0 -1/2 -1/2  1/2    0]
[   0    0    1    0  1/2]
[   0    0    0    0    1]
sage: t * S.decomposition_matrix() == 1
True

Python

>>> from sage.all import *
>>> S = ModularSymbols(Integer(37), Integer(2))
>>> t = S.decomposition_matrix_inverse(); t
[   1    0    0    0  1/3]
[   0  1/2 -1/2  1/2 -1/2]
[   0 -1/2 -1/2  1/2    0]
[   0    0    1    0  1/2]
[   0    0    0    0    1]
>>> t * S.decomposition_matrix() == Integer(1)
True

degeneracy_map(codomain, t=1)[source]¶

The \(t\)-th degeneracy map from self to the module codomain.

The level of the codomain must be a divisor or multiple of level, and \(t\) must be a divisor of the quotient.

INPUT:

codomain – a Hecke module, which should be of the same type as self, or a positive integer (in which case Sage will use hecke_module_of_level() to find the “natural” module of the corresponding level).
t – integer; the parameter of the degeneracy map, i.e., the map is related to \(f(q)\) - \(f(q^t)\)

OUTPUT: a morphism from self to codomain

EXAMPLES:

Sage

sage: M = ModularSymbols(11,sign=1)
sage: d1 = M.degeneracy_map(33); d1
Hecke module morphism degeneracy map corresponding to f(q) |--> f(q)
defined by the matrix
[ 1  0  0  0 -2 -1]
[ 0 -1  1  0  0  0]
Domain:   Modular Symbols space of dimension 2 for Gamma_0(11) of weight ...
Codomain: Modular Symbols space of dimension 6 for Gamma_0(33) of weight ...
sage: M.degeneracy_map(33, 3).matrix()
[ 3  2  0  2 -2  1]
[ 0  0 -1  1  0  0]
sage: M = ModularSymbols(33, sign=1)
sage: d2 = M.degeneracy_map(11); d2.matrix()
[ 1  0]
[ 0 -2]
[ 0  2]
[ 0  1]
[-1  0]
[-1  0]
sage: (d2*d1).matrix()
[4 0]
[0 4]

Python

>>> from sage.all import *
>>> M = ModularSymbols(Integer(11),sign=Integer(1))
>>> d1 = M.degeneracy_map(Integer(33)); d1
Hecke module morphism degeneracy map corresponding to f(q) |--> f(q)
defined by the matrix
[ 1  0  0  0 -2 -1]
[ 0 -1  1  0  0  0]
Domain:   Modular Symbols space of dimension 2 for Gamma_0(11) of weight ...
Codomain: Modular Symbols space of dimension 6 for Gamma_0(33) of weight ...
>>> M.degeneracy_map(Integer(33), Integer(3)).matrix()
[ 3  2  0  2 -2  1]
[ 0  0 -1  1  0  0]
>>> M = ModularSymbols(Integer(33), sign=Integer(1))
>>> d2 = M.degeneracy_map(Integer(11)); d2.matrix()
[ 1  0]
[ 0 -2]
[ 0  2]
[ 0  1]
[-1  0]
[-1  0]
>>> (d2*d1).matrix()
[4 0]
[0 4]

Sage

sage: M = ModularSymbols(3, 12, sign=1)
sage: M.degeneracy_map(1)
Hecke module morphism degeneracy map corresponding to f(q) |--> f(q)
defined by the matrix
[1 0]
[0 0]
[0 1]
[0 1]
[0 1]
Domain: Modular Symbols space of dimension 5 for Gamma_0(3) of weight ...
Codomain: Modular Symbols space of dimension 2 for Gamma_0(1) of weight ...

Python

>>> from sage.all import *
>>> M = ModularSymbols(Integer(3), Integer(12), sign=Integer(1))
>>> M.degeneracy_map(Integer(1))
Hecke module morphism degeneracy map corresponding to f(q) |--> f(q)
defined by the matrix
[1 0]
[0 0]
[0 1]
[0 1]
[0 1]
Domain: Modular Symbols space of dimension 5 for Gamma_0(3) of weight ...
Codomain: Modular Symbols space of dimension 2 for Gamma_0(1) of weight ...

Sage

sage: S = M.cuspidal_submodule()
sage: S.degeneracy_map(1)
Hecke module morphism defined by the matrix
[1 0]
[0 0]
[0 0]
Domain: Modular Symbols subspace of dimension 3 of Modular Symbols space ...
Codomain: Modular Symbols space of dimension 2 for Gamma_0(1) of weight ...

Python

>>> from sage.all import *
>>> S = M.cuspidal_submodule()
>>> S.degeneracy_map(Integer(1))
Hecke module morphism defined by the matrix
[1 0]
[0 0]
[0 0]
Domain: Modular Symbols subspace of dimension 3 of Modular Symbols space ...
Codomain: Modular Symbols space of dimension 2 for Gamma_0(1) of weight ...

Sage

sage: D = ModularSymbols(10,4).cuspidal_submodule().decomposition()
sage: D
[
 Modular Symbols subspace of dimension 2 of
  Modular Symbols space of dimension 10 for
   Gamma_0(10) of weight 4 with sign 0 over Rational Field,
 Modular Symbols subspace of dimension 4 of
  Modular Symbols space of dimension 10 for
   Gamma_0(10) of weight 4 with sign 0 over Rational Field
]
sage: D[1].degeneracy_map(5)
Hecke module morphism defined by the matrix
[   0    0   -1    1]
[   0  1/2  3/2   -2]
[   0   -1    1    0]
[   0 -3/4 -1/4    1]
Domain:   Modular Symbols subspace of dimension 4 of Modular Symbols space ...
Codomain: Modular Symbols space of dimension 4 for Gamma_0(5) of weight ...

Python

>>> from sage.all import *
>>> D = ModularSymbols(Integer(10),Integer(4)).cuspidal_submodule().decomposition()
>>> D
[
 Modular Symbols subspace of dimension 2 of
  Modular Symbols space of dimension 10 for
   Gamma_0(10) of weight 4 with sign 0 over Rational Field,
 Modular Symbols subspace of dimension 4 of
  Modular Symbols space of dimension 10 for
   Gamma_0(10) of weight 4 with sign 0 over Rational Field
]
>>> D[Integer(1)].degeneracy_map(Integer(5))
Hecke module morphism defined by the matrix
[   0    0   -1    1]
[   0  1/2  3/2   -2]
[   0   -1    1    0]
[   0 -3/4 -1/4    1]
Domain:   Modular Symbols subspace of dimension 4 of Modular Symbols space ...
Codomain: Modular Symbols space of dimension 4 for Gamma_0(5) of weight ...

We check for a subtle caching bug that came up in work on Issue #10453:

Sage

sage: loads(dumps(J0(33).decomposition()[0].modular_symbols()))
Modular Symbols subspace of dimension 2 of
 Modular Symbols space of dimension 9 for
  Gamma_0(33) of weight 2 with sign 0 over Rational Field

Python

>>> from sage.all import *
>>> loads(dumps(J0(Integer(33)).decomposition()[Integer(0)].modular_symbols()))
Modular Symbols subspace of dimension 2 of
 Modular Symbols space of dimension 9 for
  Gamma_0(33) of weight 2 with sign 0 over Rational Field

We check that certain absurd inputs are correctly caught:

Sage

sage: chi = kronecker_character(7)
sage: ModularSymbols(Gamma0(7), 4).degeneracy_map(ModularSymbols(chi, 4))
Traceback (most recent call last):
...
ValueError: the characters of the domain and codomain must match

Python

>>> from sage.all import *
>>> chi = kronecker_character(Integer(7))
>>> ModularSymbols(Gamma0(Integer(7)), Integer(4)).degeneracy_map(ModularSymbols(chi, Integer(4)))
Traceback (most recent call last):
...
ValueError: the characters of the domain and codomain must match

dual_free_module()[source]¶

The free module dual to self, as a submodule of the dual module of the ambient space. As this space is ambient anyway, this just returns self.free_module().

EXAMPLES:

Sage

sage: M = ModularForms(2,8); M.dual_free_module()
Vector space of dimension 3 over Rational Field
sage: M.dual_free_module() is M.free_module()
True

Python

>>> from sage.all import *
>>> M = ModularForms(Integer(2),Integer(8)); M.dual_free_module()
Vector space of dimension 3 over Rational Field
>>> M.dual_free_module() is M.free_module()
True

fcp(n, var='x')[source]¶

Return the factorization of the characteristic polynomial of the Hecke operator \(T_n\) of index \(n\) acting on this space.

INPUT:

self – Hecke module invariant under the Hecke operator of index
\(n\)
n — a positive integer.
var — variable of polynomial (default: 'x')

OUTPUT:

list – list of the pairs \((g,e)\), where \(g\) is an irreducible factor of the characteristic polynomial of \(T_n\), and \(e\) is its multiplicity.

EXAMPLES:

Sage

sage: m = ModularSymbols(23, 2, sign=1)
sage: m.fcp(2)
(x - 3) * (x^2 + x - 1)
sage: m.hecke_operator(2).charpoly('x').factor()
(x - 3) * (x^2 + x - 1)

Python

>>> from sage.all import *
>>> m = ModularSymbols(Integer(23), Integer(2), sign=Integer(1))
>>> m.fcp(Integer(2))
(x - 3) * (x^2 + x - 1)
>>> m.hecke_operator(Integer(2)).charpoly('x').factor()
(x - 3) * (x^2 + x - 1)

free_module()[source]¶

Return the free module underlying this ambient Hecke module (the forgetful functor from Hecke modules to modules over the base ring)

EXAMPLES:

Sage

sage: ModularForms(59, 2).free_module()
Vector space of dimension 6 over Rational Field

Python

>>> from sage.all import *
>>> ModularForms(Integer(59), Integer(2)).free_module()
Vector space of dimension 6 over Rational Field

hecke_bound()[source]¶

Return an integer \(B\) such that the Hecke operators \(T_n\), for \(n\leq B\), generate the full Hecke algebra as a module over the base ring. Note that we include the \(n\) with \(n\) not coprime to the level.

At present this returns an unproven guess for non-cuspidal spaces which appears to be valid for \(M_k(\Gamma_0(N))\), where \(k\) and \(N\) are the weight and level of self. (It is clearly valid for cuspidal spaces of any fixed character, as a consequence of the Sturm bound theorem.) It returns a hopelessly wrong answer for spaces of full level \(\Gamma_1\).

TODO: Get rid of this dreadful bit of code.

EXAMPLES:

Sage

sage: ModularSymbols(17, 4).hecke_bound()
15
sage: ModularSymbols(Gamma1(17), 4).hecke_bound()  # wrong!
15

Python

>>> from sage.all import *
>>> ModularSymbols(Integer(17), Integer(4)).hecke_bound()
15
>>> ModularSymbols(Gamma1(Integer(17)), Integer(4)).hecke_bound()  # wrong!
15

hecke_images(i, v)[source]¶

Return images of the \(i\)-th standard basis vector under the Hecke operators \(T_p\) for all integers in \(v\).

INPUT:

i – nonnegative integer
v – list of positive integer

OUTPUT: matrix – whose rows are the Hecke images

EXAMPLES:

Sage

sage: M = ModularSymbols(DirichletGroup(13).0, 3)
sage: M.T(2)(M.0).element()
(zeta12 + 4, 0, -1, 1)
sage: M.hecke_images(0, [1,2])
[         1          0          0          0]
[zeta12 + 4          0         -1          1]

Python

>>> from sage.all import *
>>> M = ModularSymbols(DirichletGroup(Integer(13)).gen(0), Integer(3))
>>> M.T(Integer(2))(M.gen(0)).element()
(zeta12 + 4, 0, -1, 1)
>>> M.hecke_images(Integer(0), [Integer(1),Integer(2)])
[         1          0          0          0]
[zeta12 + 4          0         -1          1]

hecke_module_of_level(level)[source]¶

Return the Hecke module corresponding to self at the given level, which should be either a divisor or a multiple of the level of self.

This raises NotImplementedError, and should be overridden in derived classes.

EXAMPLES:

Sage

sage: sage.modular.hecke.ambient_module.AmbientHeckeModule.hecke_module_of_level(ModularForms(2, 8),6)
Traceback (most recent call last):
...
NotImplementedError

Python

>>> from sage.all import *
>>> sage.modular.hecke.ambient_module.AmbientHeckeModule.hecke_module_of_level(ModularForms(Integer(2), Integer(8)),Integer(6))
Traceback (most recent call last):
...
NotImplementedError

intersection(other)[source]¶

Return the intersection of self and other, which must both lie in a common ambient space of modular symbols.

EXAMPLES:

Sage

sage: M = ModularSymbols(43, sign=1)
sage: A = M[0] + M[1]
sage: B = M[1] + M[2]
sage: A.rank(), B.rank()
(2, 3)
sage: C = A.intersection(B); C.rank()  # TODO
1

Python

>>> from sage.all import *
>>> M = ModularSymbols(Integer(43), sign=Integer(1))
>>> A = M[Integer(0)] + M[Integer(1)]
>>> B = M[Integer(1)] + M[Integer(2)]
>>> A.rank(), B.rank()
(2, 3)
>>> C = A.intersection(B); C.rank()  # TODO
1

is_ambient()[source]¶

Return True if and only if self is an ambient Hecke module.

Warning

self can only be ambient by being of type AmbientHeckeModule.

For example, decomposing a simple ambient space yields a single factor, and that factor is not considered an ambient space.

EXAMPLES:

Sage

sage: m = ModularSymbols(10)
sage: m.is_ambient()
True

Python

>>> from sage.all import *
>>> m = ModularSymbols(Integer(10))
>>> m.is_ambient()
True

Sage

sage: a = m[0]  # the unique simple factor
sage: a == m
True
sage: a.is_ambient()
False

Python

>>> from sage.all import *
>>> a = m[Integer(0)]  # the unique simple factor
>>> a == m
True
>>> a.is_ambient()
False

is_full_hecke_module(compute=True)[source]¶

Return True if this space is invariant under the action of all Hecke operators, even those that divide the level. This is always true for ambient Hecke modules, so return True.

EXAMPLES:

Sage

sage: ModularSymbols(11, 4).is_full_hecke_module()
True

Python

>>> from sage.all import *
>>> ModularSymbols(Integer(11), Integer(4)).is_full_hecke_module()
True

is_new(p=None)[source]¶

Return True if this module is entirely new.

EXAMPLES:

Sage

sage: ModularSymbols(11, 4).is_new()
False
sage: ModularSymbols(1, 12).is_new()
True

Python

>>> from sage.all import *
>>> ModularSymbols(Integer(11), Integer(4)).is_new()
False
>>> ModularSymbols(Integer(1), Integer(12)).is_new()
True

is_old(p=None)[source]¶

Return True if this module is entirely old.

EXAMPLES:

Sage

sage: ModularSymbols(22).is_old()
True
sage: ModularSymbols(3, 12).is_old()
False

Python

>>> from sage.all import *
>>> ModularSymbols(Integer(22)).is_old()
True
>>> ModularSymbols(Integer(3), Integer(12)).is_old()
False

is_submodule(V)[source]¶

Return True if and only if self is a submodule of V.

Since this is an ambient space, this returns True if and only if V is equal to self.

EXAMPLES:

Sage

sage: ModularSymbols(1, 4).is_submodule(ModularSymbols(11,4))
False
sage: ModularSymbols(11, 4).is_submodule(ModularSymbols(11,4))
True

Python

>>> from sage.all import *
>>> ModularSymbols(Integer(1), Integer(4)).is_submodule(ModularSymbols(Integer(11),Integer(4)))
False
>>> ModularSymbols(Integer(11), Integer(4)).is_submodule(ModularSymbols(Integer(11),Integer(4)))
True

linear_combination_of_basis(v)[source]¶

Given a list or vector of length equal to the dimension of self, construct the appropriate linear combination of the basis vectors of self.

EXAMPLES:

Sage

sage: ModularForms(3, 12).linear_combination_of_basis([1,0,0,0,1])
2*q + 2049*q^2 + 177147*q^3 + 4196177*q^4 + 48830556*q^5 + O(q^6)

Python

>>> from sage.all import *
>>> ModularForms(Integer(3), Integer(12)).linear_combination_of_basis([Integer(1),Integer(0),Integer(0),Integer(0),Integer(1)])
2*q + 2049*q^2 + 177147*q^3 + 4196177*q^4 + 48830556*q^5 + O(q^6)

new_submodule(p=None)[source]¶

Return the new or \(p\)-new submodule of self.

INPUT:

p – (default: None) if not None, return only the \(p\)-new submodule

OUTPUT: the new or \(p\)-new submodule of self, i.e. the intersection of the kernel of the degeneracy lowering maps to level \(N/p\) (for the given prime \(p\), or for all prime divisors of \(N\) if \(p\) is not given).

If self is cuspidal, this is a Hecke-invariant complement of the corresponding old submodule, but this may break down on Eisenstein subspaces (see the amusing example in William Stein’s book of a form which is new and old at the same time).

EXAMPLES:

Sage

sage: m = ModularSymbols(33); m.rank()
9
sage: m.new_submodule().rank()
3
sage: m.new_submodule(3).rank()
4
sage: m.new_submodule(11).rank()
8

Python

>>> from sage.all import *
>>> m = ModularSymbols(Integer(33)); m.rank()
9
>>> m.new_submodule().rank()
3
>>> m.new_submodule(Integer(3)).rank()
4
>>> m.new_submodule(Integer(11)).rank()
8

nonembedded_free_module()[source]¶

Return the free module corresponding to self as an abstract free module (rather than as a submodule of an ambient free module).

As this module is ambient anyway, this just returns self.free_module().

EXAMPLES:

Sage

sage: M = ModularSymbols(11, 2)
sage: M.nonembedded_free_module() is M.free_module()
True

Python

>>> from sage.all import *
>>> M = ModularSymbols(Integer(11), Integer(2))
>>> M.nonembedded_free_module() is M.free_module()
True

old_submodule(p=None)[source]¶

Return the old or \(p\)-old submodule of self, i.e. the sum of the images of the degeneracy maps from level \(N/p\) (for the given prime \(p\), or for all primes \(p\) dividing \(N\) if \(p\) is not given).

INPUT:

p – (default: None) if not None, return only the \(p\)-old submodule

OUTPUT: the old or \(p\)-old submodule of self

EXAMPLES:

Sage

sage: m = ModularSymbols(33); m.rank()
9
sage: m.old_submodule().rank()
7
sage: m.old_submodule(3).rank()
6
sage: m.new_submodule(11).rank()
8

Python

>>> from sage.all import *
>>> m = ModularSymbols(Integer(33)); m.rank()
9
>>> m.old_submodule().rank()
7
>>> m.old_submodule(Integer(3)).rank()
6
>>> m.new_submodule(Integer(11)).rank()
8

Sage

sage: e = DirichletGroup(16)([-1, 1])
sage: M = ModularSymbols(e, 3, sign=1); M
Modular Symbols space of dimension 4 and level 16, weight 3,
 character [-1, 1], sign 1, over Rational Field
sage: M.old_submodule()
Modular Symbols subspace of dimension 3 of
 Modular Symbols space of dimension 4 and level 16, weight 3,
  character [-1, 1], sign 1, over Rational Field

Python

>>> from sage.all import *
>>> e = DirichletGroup(Integer(16))([-Integer(1), Integer(1)])
>>> M = ModularSymbols(e, Integer(3), sign=Integer(1)); M
Modular Symbols space of dimension 4 and level 16, weight 3,
 character [-1, 1], sign 1, over Rational Field
>>> M.old_submodule()
Modular Symbols subspace of dimension 3 of
 Modular Symbols space of dimension 4 and level 16, weight 3,
  character [-1, 1], sign 1, over Rational Field

Illustrate that Issue #10664 is fixed:

Sage

sage: ModularSymbols(DirichletGroup(42)[7], 6, sign=1).old_subspace(3)
Modular Symbols subspace of dimension 0 of
 Modular Symbols space of dimension 40 and level 42, weight 6,
  character [-1, -1], sign 1, over Rational Field

Python

>>> from sage.all import *
>>> ModularSymbols(DirichletGroup(Integer(42))[Integer(7)], Integer(6), sign=Integer(1)).old_subspace(Integer(3))
Modular Symbols subspace of dimension 0 of
 Modular Symbols space of dimension 40 and level 42, weight 6,
  character [-1, -1], sign 1, over Rational Field

rank()[source]¶

Return the rank of this ambient Hecke module.

OUTPUT: integer

EXAMPLES:

Sage

sage: M = sage.modular.hecke.ambient_module.AmbientHeckeModule(QQ, 3, 11, 2); M
Generic ambient Hecke module of rank 3, level 11 and weight 2 over Rational Field
sage: M.rank()
3

Python

>>> from sage.all import *
>>> M = sage.modular.hecke.ambient_module.AmbientHeckeModule(QQ, Integer(3), Integer(11), Integer(2)); M
Generic ambient Hecke module of rank 3, level 11 and weight 2 over Rational Field
>>> M.rank()
3

submodule(M, Mdual=None, check=True)[source]¶

Return the Hecke submodule of self generated by \(M\), which may be a submodule of the free module of self, or a list of elements of self.

EXAMPLES:

Sage

sage: M = ModularForms(37, 2)
sage: A = M.submodule([M.newforms()[0].element(), M.newforms()[1].element()]); A
Modular Forms subspace of dimension 2 of
 Modular Forms space of dimension 3 for
  Congruence Subgroup Gamma0(37) of weight 2 over Rational Field

Python

>>> from sage.all import *
>>> M = ModularForms(Integer(37), Integer(2))
>>> A = M.submodule([M.newforms()[Integer(0)].element(), M.newforms()[Integer(1)].element()]); A
Modular Forms subspace of dimension 2 of
 Modular Forms space of dimension 3 for
  Congruence Subgroup Gamma0(37) of weight 2 over Rational Field

submodule_from_nonembedded_module(V, Vdual=None, check=True)[source]¶

Create a submodule of this module, from a submodule of an ambient free module of the same rank as the rank of self.

INPUT:

V – submodule of ambient free module of the same rank as the rank of self
Vdual – used to pass in dual submodule (may be None)
check – whether to check that submodule is Hecke-equivariant

OUTPUT: Hecke submodule of self

EXAMPLES:

Sage

sage: V = QQ^8
sage: ModularForms(24, 2).submodule_from_nonembedded_module(V.submodule([0]))
Modular Forms subspace of dimension 0 of
 Modular Forms space of dimension 8 for
  Congruence Subgroup Gamma0(24) of weight 2 over Rational Field

Python

>>> from sage.all import *
>>> V = QQ**Integer(8)
>>> ModularForms(Integer(24), Integer(2)).submodule_from_nonembedded_module(V.submodule([Integer(0)]))
Modular Forms subspace of dimension 0 of
 Modular Forms space of dimension 8 for
  Congruence Subgroup Gamma0(24) of weight 2 over Rational Field

submodule_generated_by_images(M)[source]¶

Return the submodule of this ambient modular symbols space generated by the images under all degeneracy maps of \(M\).

The space \(M\) must have the same weight, sign, and group or character as this ambient space.

EXAMPLES:

Sage

sage: ModularSymbols(6, 12).submodule_generated_by_images(ModularSymbols(1,12))
Modular Symbols subspace of dimension 12 of
 Modular Symbols space of dimension 22 for
  Gamma_0(6) of weight 12 with sign 0 over Rational Field

Python

>>> from sage.all import *
>>> ModularSymbols(Integer(6), Integer(12)).submodule_generated_by_images(ModularSymbols(Integer(1),Integer(12)))
Modular Symbols subspace of dimension 12 of
 Modular Symbols space of dimension 22 for
  Gamma_0(6) of weight 12 with sign 0 over Rational Field

sage.modular.hecke.ambient_module.is_AmbientHeckeModule(x)[source]¶

Return True if x is of type AmbientHeckeModule.

EXAMPLES:

Sage

sage: from sage.modular.hecke.ambient_module import is_AmbientHeckeModule
sage: is_AmbientHeckeModule(ModularSymbols(6))
doctest:warning...
DeprecationWarning: the function is_AmbientHeckeModule is deprecated;
use 'isinstance(..., AmbientHeckeModule)' instead
See https://github.com/sagemath/sage/issues/37895 for details.
True
sage: is_AmbientHeckeModule(ModularSymbols(6).cuspidal_subspace())
False
sage: is_AmbientHeckeModule(ModularForms(11))
True
sage: is_AmbientHeckeModule(BrandtModule(2, 3))
True

Python

>>> from sage.all import *
>>> from sage.modular.hecke.ambient_module import is_AmbientHeckeModule
>>> is_AmbientHeckeModule(ModularSymbols(Integer(6)))
doctest:warning...
DeprecationWarning: the function is_AmbientHeckeModule is deprecated;
use 'isinstance(..., AmbientHeckeModule)' instead
See https://github.com/sagemath/sage/issues/37895 for details.
True
>>> is_AmbientHeckeModule(ModularSymbols(Integer(6)).cuspidal_subspace())
False
>>> is_AmbientHeckeModule(ModularForms(Integer(11)))
True
>>> is_AmbientHeckeModule(BrandtModule(Integer(2), Integer(3)))
True