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)
