Ambient Hecke modules¶

class sage.modular.hecke.ambient_module.AmbientHeckeModule(base_ring, rank, level, weight)

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()

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

EXAMPLE:

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

complement()

Return the largest Hecke-stable complement of this space.

EXAMPLES:

sage: M=ModularSymbols(11,2,1)
sage: 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()
sage: 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

decomposition_matrix()

Returns 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.

EXAMPLE:

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]

decomposition_matrix_inverse()

Returns the inverse of the matrix returned by decomposition_matrix().

EXAMPLE:

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

degeneracy_map(codomain, t=1)

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 - int, 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: 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  0 -2  2  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  2  0 -2  1]
[ 0  2  0 -2  0  0]
sage: M = ModularSymbols(33,sign=1)
sage: d2 = M.degeneracy_map(11); d2.matrix()
[  1   0]
[  0 1/2]
[  0  -1]
[  0   1]
[ -1   0]
[ -1   0]
sage: (d2*d1).matrix()
[4 0]
[0 4]

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 ...

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 ...

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 ...


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

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


We check that certain absurd inputs are correctly caught:

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

dual_free_module()

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().

EXAMPLE:

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

fcp(n, var='x')

Returns 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.
• int 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: 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)

free_module()

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

EXAMPLE:

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

hecke_bound()

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.

EXAMPLE:

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

hecke_images(i, v)

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 - a list of positive integer

OUTPUT:

• matrix - whose rows are the Hecke images

EXAMPLES:

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]

hecke_module_of_level(level)

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.

EXAMPLE:

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

intersection(other)

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

EXAMPLES:

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

is_ambient()

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

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

is_full_hecke_module(compute=True)

Returns 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.

EXAMPLE:

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

is_new(p=None)

Return True if this module is entirely new.

EXAMPLE:

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

is_old(p=None)

Return True if this module is entirely old.

EXAMPLE:

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

is_submodule(V)

Returns 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.

EXAMPLE:

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

linear_combination_of_basis(v)

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

EXAMPLE:

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)

new_submodule(p=None)

Returns 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: 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

nonembedded_free_module()

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

old_submodule(p=None)

Returns 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: 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

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


Illustrate that trac 10664 is fixed:

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

rank()

Return the rank of this ambient Hecke module.

OUTPUT:

Integer

EXAMPLES:

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

submodule(M, Mdual=None, check=True)

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.

EXAMPLE:

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

submodule_from_nonembedded_module(V, Vdual=None, check=True)

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

EXAMPLE:

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

submodule_generated_by_images(M)

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: 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

sage.modular.hecke.ambient_module.is_AmbientHeckeModule(x)

Return True if x is of type AmbientHeckeModule.

EXAMPLES:

sage: from sage.modular.hecke.ambient_module import is_AmbientHeckeModule
sage: is_AmbientHeckeModule(ModularSymbols(6))
True
sage: is_AmbientHeckeModule(ModularSymbols(6).cuspidal_subspace())
False
sage: is_AmbientHeckeModule(ModularForms(11))
True
sage: is_AmbientHeckeModule(BrandtModule(2, 3))
True


Hecke modules

Next topic

Submodules of Hecke modules