def newform(self, names=None):
r"""
Return the newform that this modular abelian variety is attached to.
EXAMPLES::
sage: f = Newform('37a')
sage: A = f.abelian_variety()
sage: A.newform()
q - 2*q^2 - 3*q^3 + 2*q^4 - 2*q^5 + O(q^6)
sage: A.newform() is f
True
If the a variable name has not been specified, we must specify one::
sage: A = AbelianVariety('67b')
sage: A.newform()
Traceback (most recent call last):
...
TypeError: You must specify the name of the generator.
sage: A.newform('alpha')
q + alpha*q^2 + (-alpha - 3)*q^3 + (-3*alpha - 3)*q^4 - 3*q^5 + O(q^6)
If the eigenform is actually over `\QQ` then we don't have to specify
the name::
sage: A = AbelianVariety('67a')
sage: A.newform()
q + 2*q^2 - 2*q^3 + 2*q^4 + 2*q^5 + O(q^6)
"""
try:
return self.__named_newforms[names]
except KeyError:
self.__named_newforms[names] = Newform(self.__f.parent().change_ring(QQ), self.__f.modular_symbols(1), names=names, check=False)
return self.__named_newforms[names]
def minimal_twist(self):
r"""
Return a newform (not necessarily unique) which is a twist of the
original form `f` by a Dirichlet character of `p`-power conductor, and
which has minimal level among such twists of `f`.
An error will be raised if `f` is already minimal.
EXAMPLES::
sage: from sage.modular.local_comp.type_space import TypeSpace, example_type_space
sage: T = example_type_space(1)
sage: T.form().q_expansion(12)
q - q^2 + 2*q^3 + q^4 - 2*q^6 - q^8 + q^9 + O(q^12)
sage: g = T.minimal_twist()
sage: g.q_expansion(12)
q - q^2 - 2*q^3 + q^4 + 2*q^6 + q^7 - q^8 + q^9 + O(q^12)
sage: g.level()
14
sage: TypeSpace(g, 7).is_minimal()
True
Test that :trac:`13158` is fixed::
sage: f = Newforms(256,names='a')[0]
sage: T = TypeSpace(f,2)
sage: g = T.minimal_twist()
sage: g[0:3]
[0, 1, 0]
sage: str(g[3]) in ('a', '-a', '-1/2*a', '1/2*a')
True
sage: g[4:]
[]
sage: g.level()
64
"""
if self.is_minimal():
raise ValueError("Form is already minimal")
NN = self.form().level()
V = self.t_space
A = V.ambient()
while not V.is_submodule(A.new_submodule()):
NN = NN / self.prime()
D1 = A.degeneracy_map(NN, 1)
Dp = A.degeneracy_map(NN, self.prime())
A = D1.codomain()
vecs = [D1(v).element()
for v in V.basis()] + [Dp(v).element() for v in V.basis()]
VV = A.free_module().submodule(vecs)
V = A.submodule(VV, check=False)
D = V.decomposition()[0]
#if len(D.star_eigenvalues()) == 2:
# D = D.sign_submodule(1)
D1 = D.modular_symbols_of_sign(1)
M = ModularForms(D1.group(), D1.weight(), D1.base_ring())
ff = Newform(M, D1, names='a')
return ff