def _hecke_tp_charpoly(self, p, var='x', algorithm='linbox'):
a = p ** (self.wt - 2) + 1
N = self.klingeneisensteinAndCuspForms()
S = CuspForms(1, self.wt)
m = S.dimension()
R = PolynomialRing(QQ, names=var)
x = R.gens()[0]
f = R(S.hecke_matrix(p).charpoly(var=var, algorithm=algorithm))
f1 = f.subs({x: a ** (-1) * x}) * a ** m
g = R(N.hecke_matrix(p).charpoly(var=var, algorithm=algorithm))
return R(g / f1)
def _hecke_tp2_charpoly(self, p, var='x', algorithm='linbox'):
u = p ** (self.wt - 2)
N = self.klingeneisensteinAndCuspForms()
S = CuspForms(1, self.wt)
m = S.dimension()
R = PolynomialRing(QQ, names=var)
x = R.gens()[0]
f = R(S.hecke_matrix(p).charpoly(var=var, algorithm=algorithm))
g = R(N.hecke_matrix(p ** 2).charpoly(var=var, algorithm=algorithm))
def morph(a, b, f, m):
G = (-1) ** m * f.subs({x: -x}) * f
alst = [[k // 2, v] for k, v in G.dict().iteritems()]
F = sum([v * x ** k for k, v in alst])
return a ** m * F.subs({x: (x - b) / a})
f1 = morph(u ** 2 + u + 1, -p * u ** 3 - u ** 2 - p * u, f, m)
return R(g / f1)
def compare_vv_scalar(V, k):
dv = V.dimension_cusp_forms(k)
N = V._level
m = V._M.order()
if k in ZZ:
D = DirichletGroup(N)
if is_even(k):
chi = D(kronecker_character(m))
else:
chi = D(kronecker_character(-m))
S = CuspForms(chi, k)
N = Newforms(chi, k, names='a')
ds = S.dimension()
B = N
else:
D = magma.DirichletGroup(V._level)
chi = magma.D(magma.KroneckerCharacter(2 * m))
M = magma.HalfIntegralWeightForms(chi, k)
S = M.CuspidalSubspace()
ds = S.Dimension()
B = S.Basis()
return dv, ds, S, B
def dimension_of_nolift_cuspforms(self):
if self.wt % 2 == 1:
return self.dimension()
S = CuspForms(1, (self.wt - 1) * 2)
return self.dimension_of_cuspforms() - S.dimension()
def dimension_of_cuspforms(self):
if self.wt % 2 == 1:
return self.dimension()
S = CuspForms(1, self.wt)
return self.dimension() - S.dimension()
