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# Elements of Hecke modules

AUTHORS:

- William Stein

class `sage.modular.hecke.element.``HeckeModuleElement`(parent, x=None)

  Bases: `sage.structure.element.ModuleElement`

  Element of a Hecke module.

  `ambient_module`()

    Return the ambient Hecke module that contains this element.

    EXAMPLES:

        sage: BrandtModule(37)([0,1,-1]).ambient_module()
        Brandt module of dimension 3 of level 37 of weight 2 over Rational Field

  `element`()

    Return underlying vector space element that defines this Hecke module element.

    EXAMPLES:

        sage: z = BrandtModule(37)([0,1,-1]).element(); z
        (0, 1, -1)
        sage: type(z)
        <type 'sage.modules.vector_rational_dense.Vector_rational_dense'>

  `is_cuspidal`()

    Return True if this element is cuspidal.

    EXAMPLES:

        sage: M = ModularForms(2, 22); M.0.is_cuspidal()
        True
        sage: (M.0 + M.4).is_cuspidal()
        False
        sage: EllipticCurve('37a1').newform().is_cuspidal()
        True

    It works for modular symbols too:

        sage: M = ModularSymbols(19,2)
        sage: M.0.is_cuspidal()
        False
        sage: M.1.is_cuspidal()
        True

  `is_eisenstein`()

    Return True if this element is Eisenstein. This makes sense for both modular forms and modular symbols.

    EXAMPLES:

        sage: CuspForms(2,8).0.is_eisenstein()
        False
        sage: M = ModularForms(2,8);(M.0  + M.1).is_eisenstein()
        False
        sage: M.1.is_eisenstein()
        True
        sage: ModularSymbols(19,4).0.is_eisenstein()
        False
        sage: EllipticCurve('37a1').newform().is_eisenstein()
        False

  `is_new`(p=None)

    Return True if this element is p-new. If p is None, return True if the element is new.

    EXAMPLE:

        sage: CuspForms(22, 2).0.is_new(2)
        False
        sage: CuspForms(22, 2).0.is_new(11)
        True
        sage: CuspForms(22, 2).0.is_new()
        False

  `is_old`(p=None)

    Return True if this element is p-old. If p is None, return True if the element is old.

    EXAMPLE:

        sage: CuspForms(22, 2).0.is_old(11)
        False
        sage: CuspForms(22, 2).0.is_old(2)
        True
        sage: CuspForms(22, 2).0.is_old()
        True
        sage: EisensteinForms(144, 2).1.is_old()  # long time (3s on sage.math, 2011)
        False
        sage: EisensteinForms(144, 2).1.is_old(2) # not implemented
        False

`sage.modular.hecke.element.``is_HeckeModuleElement`(x)

  Return True if x is a Hecke module element, i.e., of type HeckeModuleElement.

  EXAMPLES:

      sage: sage.modular.hecke.all.is_HeckeModuleElement(0)
      False
      sage: sage.modular.hecke.all.is_HeckeModuleElement(BrandtModule(37)([1,2,3]))
      True

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