Elements of Hecke modules¶
AUTHORS:
William Stein
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class
sage.modular.hecke.element.
HeckeModuleElement
(parent, x=None)¶ Bases:
sage.structure.element.ModuleElement
Element of a Hecke module.
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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
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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'>
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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
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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
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is_new
(p=None)¶ Return True if this element is p-new. If p is None, return True if the element is new.
EXAMPLES:
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
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is_old
(p=None)¶ Return True if this element is p-old. If p is None, return True if the element is old.
EXAMPLES:
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
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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