# Morphisms of Hecke modules¶

AUTHORS:

• William Stein
class sage.modular.hecke.morphism.HeckeModuleMorphism

Abstract base class for morphisms of Hecke modules.

class sage.modular.hecke.morphism.HeckeModuleMorphism_matrix(parent, A, name='')

Morphisms of Hecke modules when the morphism is given by a matrix.

Note that care is needed when composing morphisms, because morphisms in Sage act on the left, but their matrices act on the right (!). So if F: A -> B and G : B -> C are morphisms, the composition A -> C is G*F, but its matrix is F.matrix() * G.matrix().

EXAMPLE:

sage: A = ModularForms(1, 4)
sage: B = ModularForms(1, 16)
sage: C = ModularForms(1, 28)
sage: F = A.Hom(B)(matrix(QQ,1,2,srange(1, 3)))
sage: G = B.Hom(C)(matrix(QQ,2,3,srange(1, 7)))
sage: G * F
Hecke module morphism defined by the matrix
[ 9 12 15]
Domain: Modular Forms space of dimension 1 for Modular Group SL(2,Z) ...
Codomain: Modular Forms space of dimension 3 for Modular Group SL(2,Z) ...
sage: F * G
Traceback (most recent call last):
...
TypeError: Incompatible composition of morphisms: domain of left morphism must be codomain of right.

name(new=None)

Return the name of this operator, or set it to a new name.

EXAMPLES:

sage: M = ModularSymbols(6)
sage: t = M.Hom(M)(matrix(QQ,3,3,srange(9)), name="spam"); t
Hecke module morphism spam defined by ...
sage: t.name()
'spam'
sage: t.name("eggs"); t
Hecke module morphism eggs defined by ...

sage.modular.hecke.morphism.is_HeckeModuleMorphism(x)

Return True if x is of type HeckeModuleMorphism.

EXAMPLES:

sage: sage.modular.hecke.morphism.is_HeckeModuleMorphism(ModularSymbols(6).hecke_operator(7).hecke_module_morphism())
True

sage.modular.hecke.morphism.is_HeckeModuleMorphism_matrix(x)

EXAMPLES:

sage: sage.modular.hecke.morphism.is_HeckeModuleMorphism_matrix(ModularSymbols(6).hecke_operator(7).matrix_form().hecke_module_morphism())
True


#### Previous topic

Hom spaces between Hecke modules

Degeneracy maps