def characters(self):
r"""
Return the two characters `(\chi_1, \chi_2)` such this representation
`\pi_{f, p}` is equal to the principal series `\pi(\chi_1, \chi_2)`.
These are the unramified characters mapping `p` to the roots of the Satake polynomial,
so in most cases (but not always) they will be defined over an
extension of the coefficient field of self.
EXAMPLES::
sage: LocalComponent(Newform('11a'), 17).characters()
[
Character of Q_17*, of level 0, mapping 17 |--> d,
Character of Q_17*, of level 0, mapping 17 |--> -d - 2
]
sage: LocalComponent(Newforms(Gamma1(5), 6, names='a')[1], 3).characters()
[
Character of Q_3*, of level 0, mapping 3 |--> -3/2*a1 + 12,
Character of Q_3*, of level 0, mapping 3 |--> -3/2*a1 - 12
]
"""
f = self.satake_polynomial()
if not f.is_irreducible():
# This can happen; see the second example above
d = f.roots()[0][0]
else:
d = self.coefficient_field().extension(f, 'd').gen()
G = SmoothCharacterGroupQp(self.prime(), d.parent())
return Sequence([
G.character(0, [d]),
G.character(0, [self.newform()[self.prime()] - d])
],
cr=True,
universe=G)
def characters(self):
r"""
Return the two characters `(\chi_1, \chi_2)` such this representation
`\pi_{f, p}` is equal to the principal series `\pi(\chi_1, \chi_2)`.
These are the unramified characters mapping `p` to the roots of the Satake polynomial,
so in most cases (but not always) they will be defined over an
extension of the coefficient field of self.
EXAMPLES::
sage: LocalComponent(Newform('11a'), 17).characters()
[
Character of Q_17*, of level 0, mapping 17 |--> d,
Character of Q_17*, of level 0, mapping 17 |--> -d - 2
]
sage: LocalComponent(Newforms(Gamma1(5), 6, names='a')[1], 3).characters()
[
Character of Q_3*, of level 0, mapping 3 |--> -3/2*a1 + 12,
Character of Q_3*, of level 0, mapping 3 |--> -3/2*a1 - 12
]
"""
f = self.satake_polynomial()
if not f.is_irreducible():
# This can happen; see the second example above
d = f.roots()[0][0]
else:
d = self.coefficient_field().extension(f, 'd').gen()
G = SmoothCharacterGroupQp(self.prime(), d.parent())
return Sequence([G.character(0, [d]), G.character(0, [self.newform()[self.prime()] - d])], cr=True, universe=G)
def characters(self):
r"""
Return the two characters `(\chi_1, \chi_2)` such that the local component `\pi_{f, p}` is the induction of the character `\chi_1 \times \chi_2` of the Borel subgroup.
EXAMPLE::
sage: LocalComponent(Newforms(Gamma1(13), 2, names='a')[0], 13).characters()
[
Character of Q_13*, of level 0, mapping 13 |--> 3*a0 + 2,
Character of Q_13*, of level 1, mapping 2 |--> a0 + 2, 13 |--> -3*a0 - 7
]
"""
G = SmoothCharacterGroupQp(self.prime(), self.coefficient_field())
chi1 = G.character(0, [self.newform()[self.prime()]])
chi2 = G.character(0, [self.prime()]) * self.central_character() / chi1
return Sequence([chi1, chi2], cr=True, universe=G)
def characters(self):
r"""
Return the two characters `(\chi_1, \chi_2)` such that the local component `\pi_{f, p}` is the induction of the character `\chi_1 \times \chi_2` of the Borel subgroup.
EXAMPLE::
sage: LocalComponent(Newforms(Gamma1(13), 2, names='a')[0], 13).characters()
[
Character of Q_13*, of level 0, mapping 13 |--> 3*a0 + 2,
Character of Q_13*, of level 1, mapping 2 |--> a0 + 2, 13 |--> -3*a0 - 7
]
"""
G = SmoothCharacterGroupQp(self.prime(), self.coefficient_field())
chi1 = G.character(0, [self.newform()[self.prime()]])
chi2 = G.character(0, [self.prime()]) * self.central_character() / chi1
return Sequence([chi1, chi2], cr=True, universe=G)
def central_character(self):
r"""
Return the central character of this representation. This is the
restriction to `\QQ_p^\times` of the unique smooth character `\omega`
of `\mathbf{A}^\times / \QQ^\times` such that `\omega(\varpi_\ell) =
\ell^j \varepsilon(\ell)` for all primes `\ell \nmid Np`, where
`\varpi_\ell` is a uniformiser at `\ell`, `\varepsilon` is the
Nebentypus character of the newform `f`, and `j` is the twist factor
(see the documentation for :func:`~LocalComponent`).
EXAMPLES::
sage: LocalComponent(Newform('27a'), 3).central_character()
Character of Q_3*, of level 0, mapping 3 |--> 1
sage: LocalComponent(Newforms(Gamma1(5), 5, names='c')[0], 5).central_character()
Character of Q_5*, of level 1, mapping 2 |--> c0 + 1, 5 |--> 125
sage: LocalComponent(Newforms(DirichletGroup(24)([1, -1,-1]), 3, names='a')[0], 2).central_character()
Character of Q_2*, of level 3, mapping 7 |--> 1, 5 |--> -1, 2 |--> -2
"""
from sage.arith.all import crt
chi = self.newform().character()
f = self.prime()**self.conductor()
N = self.newform().level() // f
G = DirichletGroup(f, self.coefficient_field())
chip = G([chi(crt(ZZ(x), 1, f, N))
for x in G.unit_gens()]).primitive_character()
a = crt(1, self.prime(), f, N)
if chip.conductor() == 1:
return SmoothCharacterGroupQp(
self.prime(), self.coefficient_field()).character(
0, [chi(a) * self.prime()**self.twist_factor()])
else:
return SmoothCharacterGroupQp(
self.prime(), self.coefficient_field()).character(
chip.conductor().valuation(self.prime()),
list((~chip).values_on_gens()) +
[chi(a) * self.prime()**self.twist_factor()])
def characters(self):
r"""
Return the defining characters of this representation. In this case, it
will return the unique unramified character `\chi` of `\QQ_p^\times`
such that this representation is equal to `\mathrm{St} \otimes \chi`,
where `\mathrm{St}` is the Steinberg representation (defined as the
quotient of the parabolic induction of the trivial character by its
trivial subrepresentation).
EXAMPLES:
Our first example is the newform corresponding to an elliptic curve of
conductor `37`. This is the nontrivial quadratic twist of Steinberg,
corresponding to the fact that the elliptic curve has non-split
multiplicative reduction at 37::
sage: LocalComponent(Newform('37a'), 37).characters()
[Character of Q_37*, of level 0, mapping 37 |--> -1]
We try an example in odd weight, where the central character isn't
trivial::
sage: Pi = LocalComponent(Newforms(DirichletGroup(21)([-1, 1]), 3, names='j')[0], 7); Pi.characters()
[Character of Q_7*, of level 0, mapping 7 |--> -1/2*j0^2 - 7/2]
sage: Pi.characters()[0] ^2 == Pi.central_character()
True
An example using a non-standard twist factor::
sage: Pi = LocalComponent(Newforms(DirichletGroup(21)([-1, 1]), 3, names='j')[0], 7, twist_factor=3); Pi.characters()
[Character of Q_7*, of level 0, mapping 7 |--> -7/2*j0^2 - 49/2]
sage: Pi.characters()[0]^2 == Pi.central_character()
True
"""
return [
SmoothCharacterGroupQp(
self.prime(),
self.coefficient_field()).character(0, [
self.newform()[self.prime()] * self.prime()
**((self.twist_factor() - self.newform().weight() + 2) / 2)
])
]
