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.rings.arith 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 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 get_magma_qexpansions(filename, i1, prec, base_ring, magma=None):
if magma is None:
from sage.interfaces.magma import Magma
magma = Magma()
magma.set('prec',prec)
magma.load("get_qexpansions.m")
if i1 is None:
for line in filename:
print line
magma.eval(line)
magma.eval("f := g") # In case text is reversed
f = 'f'
eps_data_f = 'eps_on_gens'
else:
magma.load(filename)
f = 'eigenforms_list[%s][1]'%i1
eps_data_f = 'eigenforms_list[%s][2]'%i1
qexpm = magma.extend_qexpansion(f, eps_data_f, prec)
F0 = [0] * qexpm.Valuation().sage() + [o.sage() for o in qexpm.ElementToSequence()]
K = F0[-1].parent()
a = K.gen()
phi = find_embeddings(K,base_ring)[0]
F = [phi(o) for o in F0]
eps_f = magma.get_character(f, eps_data_f).ValueList().sage()
eps_f_full = magma.get_character_full(f, eps_data_f).ValueList().sage()
N = len(eps_f)
Geps = DirichletGroup(N, base_ring = K)
eps_f = Geps([eps_f[i - 1] for i in Geps.unit_gens()])
Geps_full = DirichletGroup(N)
psi = eps_f_full[0].parent().embeddings(Geps_full.base_ring())[0]
eps_f_full = Geps_full([psi(eps_f_full[i - 1]) for i in Geps_full.unit_gens()])
if a == 1:
sigma = lambda x:x
else:
try:
sigma = next((s for s in K.automorphisms() if s(a)*a == 1))
except StopIteration:
raise NotImplementedError
G = [phi(sigma(o)) for o in F0]
eps_g = eps_f**-1
eps_g_full = eps_f_full**-1
F = ModFormqExp(F, base_ring, weight=1, level = N, character = eps_f, character_full = eps_f_full)
G = ModFormqExp(G, base_ring, weight=1, level = N, character = eps_g, character_full = eps_g_full)
return F, G
def dimension_of_ordinary_subspace(self, p=None, cusp=False):
"""
If ``cusp`` is ``True``, return dimension of cuspidal ordinary
subspace. This does a weight 2 computation with sage's ModularSymbols.
EXAMPLES::
sage: M = OverconvergentModularSymbols(11, 0, sign=-1, p=3, prec_cap=4, base=ZpCA(3, 8))
sage: M.dimension_of_ordinary_subspace()
2
sage: M.dimension_of_ordinary_subspace(cusp=True)
2
sage: M = OverconvergentModularSymbols(11, 0, sign=1, p=3, prec_cap=4, base=ZpCA(3, 8))
sage: M.dimension_of_ordinary_subspace(cusp=True)
2
sage: M.dimension_of_ordinary_subspace()
4
sage: M = OverconvergentModularSymbols(11, 0, sign=0, p=3, prec_cap=4, base=ZpCA(3, 8))
sage: M.dimension_of_ordinary_subspace()
6
sage: M.dimension_of_ordinary_subspace(cusp=True)
4
sage: M = OverconvergentModularSymbols(11, 0, sign=1, p=11, prec_cap=4, base=ZpCA(11, 8))
sage: M.dimension_of_ordinary_subspace(cusp=True)
1
sage: M.dimension_of_ordinary_subspace()
2
sage: M = OverconvergentModularSymbols(11, 2, sign=1, p=11, prec_cap=4, base=ZpCA(11, 8))
sage: M.dimension_of_ordinary_subspace(cusp=True)
0
sage: M.dimension_of_ordinary_subspace()
1
sage: M = OverconvergentModularSymbols(11, 10, sign=1, p=11, prec_cap=4, base=ZpCA(11, 8))
sage: M.dimension_of_ordinary_subspace(cusp=True)
1
sage: M.dimension_of_ordinary_subspace()
2
An example with odd weight and hence non-trivial character::
sage: K = Qp(11, 6)
sage: DG = DirichletGroup(11, K)
sage: chi = DG([K(378703)])
sage: MM = FamiliesOfOMS(chi, 1, p=11, prec_cap=[4, 4], base_coeffs=ZpCA(11, 4), sign=-1)
sage: MM.dimension_of_ordinary_subspace()
1
"""
try:
p = self.prime()
except AttributeError:
if p is None:
raise ValueError(
"If self doesn't have a prime, must specify p.")
try:
return self._ord_dim_dict[(p, cusp)]
except AttributeError:
self._ord_dim_dict = {}
except KeyError:
pass
from sage.modular.dirichlet import DirichletGroup
from sage.rings.finite_rings.constructor import GF
try:
chi = self.character()
except AttributeError:
chi = DirichletGroup(self.level(), GF(p))[0]
if chi is None:
chi = DirichletGroup(self.level(), GF(p))[0]
from sage.modular.modsym.modsym import ModularSymbols
r = self.weight() % (p - 1)
if chi.is_trivial():
N = chi.modulus()
if N % p != 0:
N *= p
else:
e = N.valuation(p)
N.divide_knowing_divisible_by(p**(e - 1))
chi = DirichletGroup(N, GF(p))[0]
elif chi.modulus() % p != 0:
chi = DirichletGroup(chi.modulus() * p, GF(p))(chi)
DG = DirichletGroup(chi.modulus(), GF(p))
if r == 0:
from sage.modular.arithgroup.congroup_gamma0 import Gamma0_constructor as Gamma0
verbose("in dim: %s, %s, %s" % (self.sign(), chi, p))
M = ModularSymbols(DG(chi), 2, self.sign(), GF(p))
else:
psi = [GF(p)(u)**r for u in DG.unit_gens()] #mod p Teichmuller^r
psi = DG(psi)
M = ModularSymbols(DG(chi) * psi, 2, self.sign(), GF(p))
if cusp:
M = M.cuspidal_subspace()
hecke_poly = M.hecke_polynomial(p)
verbose("in dim: %s" % (hecke_poly))
x = hecke_poly.parent().gen()
d = hecke_poly.degree() - hecke_poly.ord(x)
self._ord_dim_dict[(p, cusp)] = d
return d
def dimension_of_ordinary_subspace(self, p=None, cusp=False):
"""
If ``cusp`` is ``True``, return dimension of cuspidal ordinary
subspace. This does a weight 2 computation with sage's ModularSymbols.
EXAMPLES::
sage: M = OverconvergentModularSymbols(11, 0, sign=-1, p=3, prec_cap=4, base=ZpCA(3, 8))
sage: M.dimension_of_ordinary_subspace()
2
sage: M.dimension_of_ordinary_subspace(cusp=True)
2
sage: M = OverconvergentModularSymbols(11, 0, sign=1, p=3, prec_cap=4, base=ZpCA(3, 8))
sage: M.dimension_of_ordinary_subspace(cusp=True)
2
sage: M.dimension_of_ordinary_subspace()
4
sage: M = OverconvergentModularSymbols(11, 0, sign=0, p=3, prec_cap=4, base=ZpCA(3, 8))
sage: M.dimension_of_ordinary_subspace()
6
sage: M.dimension_of_ordinary_subspace(cusp=True)
4
sage: M = OverconvergentModularSymbols(11, 0, sign=1, p=11, prec_cap=4, base=ZpCA(11, 8))
sage: M.dimension_of_ordinary_subspace(cusp=True)
1
sage: M.dimension_of_ordinary_subspace()
2
sage: M = OverconvergentModularSymbols(11, 2, sign=1, p=11, prec_cap=4, base=ZpCA(11, 8))
sage: M.dimension_of_ordinary_subspace(cusp=True)
0
sage: M.dimension_of_ordinary_subspace()
1
sage: M = OverconvergentModularSymbols(11, 10, sign=1, p=11, prec_cap=4, base=ZpCA(11, 8))
sage: M.dimension_of_ordinary_subspace(cusp=True)
1
sage: M.dimension_of_ordinary_subspace()
2
An example with odd weight and hence non-trivial character::
sage: K = Qp(11, 6)
sage: DG = DirichletGroup(11, K)
sage: chi = DG([K(378703)])
sage: MM = FamiliesOfOMS(chi, 1, p=11, prec_cap=[4, 4], base_coeffs=ZpCA(11, 4), sign=-1)
sage: MM.dimension_of_ordinary_subspace()
1
"""
try:
p = self.prime()
except AttributeError:
if p is None:
raise ValueError("If self doesn't have a prime, must specify p.")
try:
return self._ord_dim_dict[(p, cusp)]
except AttributeError:
self._ord_dim_dict = {}
except KeyError:
pass
from sage.modular.dirichlet import DirichletGroup
from sage.rings.finite_rings.constructor import GF
try:
chi = self.character()
except AttributeError:
chi = DirichletGroup(self.level(), GF(p))[0]
if chi is None:
chi = DirichletGroup(self.level(), GF(p))[0]
from sage.modular.modsym.modsym import ModularSymbols
r = self.weight() % (p-1)
if chi.is_trivial():
N = chi.modulus()
if N % p != 0:
N *= p
else:
e = N.valuation(p)
N.divide_knowing_divisible_by(p ** (e-1))
chi = DirichletGroup(N, GF(p))[0]
elif chi.modulus() % p != 0:
chi = DirichletGroup(chi.modulus() * p, GF(p))(chi)
DG = DirichletGroup(chi.modulus(), GF(p))
if r == 0:
from sage.modular.arithgroup.congroup_gamma0 import Gamma0_constructor as Gamma0
verbose("in dim: %s, %s, %s"%(self.sign(), chi, p))
M = ModularSymbols(DG(chi), 2, self.sign(), GF(p))
else:
psi = [GF(p)(u) ** r for u in DG.unit_gens()] #mod p Teichmuller^r
psi = DG(psi)
M = ModularSymbols(DG(chi) * psi, 2, self.sign(), GF(p))
if cusp:
M = M.cuspidal_subspace()
hecke_poly = M.hecke_polynomial(p)
verbose("in dim: %s"%(hecke_poly))
x = hecke_poly.parent().gen()
d = hecke_poly.degree() - hecke_poly.ord(x)
self._ord_dim_dict[(p, cusp)] = d
return d