def __call__(self, x, check=True):
r"""
Convert x into an element of this Hecke algebra. Here x is either:
- an element of a Hecke algebra equal to this one
- an element of the corresponding anemic Hecke algebra, if x is a full
Hecke algebra
- an element of the corresponding full Hecke algebra of the
form `T_i` where i is coprime to ``self.level()``, if self
is an anemic Hecke algebra
- something that can be converted into an element of the
underlying matrix space.
In the last case, the parameter ``check'' controls whether or
not to check that this element really does lie in the
appropriate algebra. At present, setting ``check=True'' raises
a NotImplementedError unless x is a scalar (or a diagonal
matrix).
EXAMPLES::
sage: T = ModularSymbols(11).hecke_algebra()
sage: T.gen(2) in T
True
sage: 5 in T
True
sage: T.gen(2).matrix() in T
Traceback (most recent call last):
...
NotImplementedError: Membership testing for '...' not implemented
sage: T(T.gen(2).matrix(), check=False)
Hecke operator on Modular Symbols space of dimension 3 for Gamma_0(11) of weight 2 with sign 0 over Rational Field defined by:
[ 3 0 -1]
[ 0 -2 0]
[ 0 0 -2]
sage: A = ModularSymbols(11).anemic_hecke_algebra()
sage: A(T.gen(3))
Hecke operator T_3 on Modular Symbols space of dimension 3 for Gamma_0(11) of weight 2 with sign 0 over Rational Field
sage: A(T.gen(11))
Traceback (most recent call last):
...
TypeError: Don't know how to construct an element of Anemic Hecke algebra acting on Modular Symbols space of dimension 3 for Gamma_0(11) of weight 2 with sign 0 over Rational Field from Hecke operator T_11 on Modular Symbols space of dimension 3 for Gamma_0(11) of weight 2 with sign 0 over Rational Field
TESTS:
We test that coercion is OK between the Hecke algebras associated to two submodules which are equal but have different bases::
sage: M = CuspForms(Gamma0(57))
sage: f1,f2,f3 = M.newforms()
sage: N1 = M.submodule(M.free_module().submodule_with_basis([f1.element().element(), f2.element().element()]))
sage: N2 = M.submodule(M.free_module().submodule_with_basis([f1.element().element(), (f1.element() + f2.element()).element()]))
sage: N1.hecke_operator(5).matrix_form()
Hecke operator on Modular Forms subspace of dimension 2 of ... defined by:
[-3 0]
[ 0 1]
sage: N2.hecke_operator(5).matrix_form()
Hecke operator on Modular Forms subspace of dimension 2 of ... defined by:
[-3 0]
[-4 1]
sage: N1.hecke_algebra()(N2.hecke_operator(5)).matrix_form()
Hecke operator on Modular Forms subspace of dimension 2 of ... defined by:
[-3 0]
[ 0 1]
sage: N1.hecke_algebra()(N2.hecke_operator(5).matrix_form())
Hecke operator on Modular Forms subspace of dimension 2 of ... defined by:
[-3 0]
[ 0 1]
"""
try:
if not isinstance(x, Element):
x = self.base_ring()(x)
if x.parent() is self:
return x
elif hecke_operator.is_HeckeOperator(x):
if x.parent() == self \
or (self.is_anemic() == False and x.parent() == self.anemic_subalgebra()) \
or (self.is_anemic() == True and x.parent().anemic_subalgebra() == self and arith.gcd(x.index(), self.level()) == 1):
return hecke_operator.HeckeOperator(self, x.index())
else:
raise TypeError
elif hecke_operator.is_HeckeAlgebraElement(x):
if x.parent() == self or (self.is_anemic() == False
and x.parent()
== self.anemic_subalgebra()):
if x.parent().module().basis_matrix() == self.module(
).basis_matrix():
return hecke_operator.HeckeAlgebraElement_matrix(
self, x.matrix())
else:
A = matrix([self.module().coordinate_vector(x.parent().module().gen(i)) \
for i in xrange(x.parent().module().rank())])
return hecke_operator.HeckeAlgebraElement_matrix(
self, ~A * x.matrix() * A)
elif x.parent() == self.anemic_subalgebra():
pass
else:
raise TypeError
else:
A = self.matrix_space()(x)
if check:
if not A.is_scalar():
raise NotImplementedError, "Membership testing for '%s' not implemented" % self
return hecke_operator.HeckeAlgebraElement_matrix(self, A)
except TypeError:
raise TypeError, "Don't know how to construct an element of %s from %s" % (
self, x)
def __call__(self, x, check=True):
r"""
Convert x into an element of this Hecke algebra. Here x is either:
- an element of a Hecke algebra equal to this one
- an element of the corresponding anemic Hecke algebra, if x is a full
Hecke algebra
- an element of the corresponding full Hecke algebra of the
form `T_i` where i is coprime to ``self.level()``, if self
is an anemic Hecke algebra
- something that can be converted into an element of the
underlying matrix space.
In the last case, the parameter ``check'' controls whether or
not to check that this element really does lie in the
appropriate algebra. At present, setting ``check=True'' raises
a NotImplementedError unless x is a scalar (or a diagonal
matrix).
EXAMPLES::
sage: T = ModularSymbols(11).hecke_algebra()
sage: T.gen(2) in T
True
sage: 5 in T
True
sage: T.gen(2).matrix() in T
Traceback (most recent call last):
...
NotImplementedError: Membership testing for '...' not implemented
sage: T(T.gen(2).matrix(), check=False)
Hecke operator on Modular Symbols space of dimension 3 for Gamma_0(11) of weight 2 with sign 0 over Rational Field defined by:
[ 3 0 -1]
[ 0 -2 0]
[ 0 0 -2]
sage: A = ModularSymbols(11).anemic_hecke_algebra()
sage: A(T.gen(3))
Hecke operator T_3 on Modular Symbols space of dimension 3 for Gamma_0(11) of weight 2 with sign 0 over Rational Field
sage: A(T.gen(11))
Traceback (most recent call last):
...
TypeError: Don't know how to construct an element of Anemic Hecke algebra acting on Modular Symbols space of dimension 3 for Gamma_0(11) of weight 2 with sign 0 over Rational Field from Hecke operator T_11 on Modular Symbols space of dimension 3 for Gamma_0(11) of weight 2 with sign 0 over Rational Field
TESTS:
We test that coercion is OK between the Hecke algebras associated to two submodules which are equal but have different bases::
sage: M = CuspForms(Gamma0(57))
sage: f1,f2,f3 = M.newforms()
sage: N1 = M.submodule(M.free_module().submodule_with_basis([f1.element().element(), f2.element().element()]))
sage: N2 = M.submodule(M.free_module().submodule_with_basis([f1.element().element(), (f1.element() + f2.element()).element()]))
sage: N1.hecke_operator(5).matrix_form()
Hecke operator on Modular Forms subspace of dimension 2 of ... defined by:
[-3 0]
[ 0 1]
sage: N2.hecke_operator(5).matrix_form()
Hecke operator on Modular Forms subspace of dimension 2 of ... defined by:
[-3 0]
[-4 1]
sage: N1.hecke_algebra()(N2.hecke_operator(5)).matrix_form()
Hecke operator on Modular Forms subspace of dimension 2 of ... defined by:
[-3 0]
[ 0 1]
sage: N1.hecke_algebra()(N2.hecke_operator(5).matrix_form())
Hecke operator on Modular Forms subspace of dimension 2 of ... defined by:
[-3 0]
[ 0 1]
"""
try:
if not isinstance(x, Element):
x = self.base_ring()(x)
if x.parent() is self:
return x
elif hecke_operator.is_HeckeOperator(x):
if x.parent() == self \
or (self.is_anemic() == False and x.parent() == self.anemic_subalgebra()) \
or (self.is_anemic() == True and x.parent().anemic_subalgebra() == self and arith.gcd(x.index(), self.level()) == 1):
return hecke_operator.HeckeOperator(self, x.index())
else:
raise TypeError
elif hecke_operator.is_HeckeAlgebraElement(x):
if x.parent() == self or (self.is_anemic() == False and x.parent() == self.anemic_subalgebra()):
if x.parent().module().basis_matrix() == self.module().basis_matrix():
return hecke_operator.HeckeAlgebraElement_matrix(self, x.matrix())
else:
A = matrix([self.module().coordinate_vector(x.parent().module().gen(i)) \
for i in xrange(x.parent().module().rank())])
return hecke_operator.HeckeAlgebraElement_matrix(self, ~A * x.matrix() * A)
elif x.parent() == self.anemic_subalgebra():
pass
else:
raise TypeError
else:
A = self.matrix_space()(x)
if check:
if not A.is_scalar():
raise NotImplementedError("Membership testing for '%s' not implemented" % self)
return hecke_operator.HeckeAlgebraElement_matrix(self, A)
except TypeError:
raise TypeError("Don't know how to construct an element of %s from %s" % (self, x))