def _compute_hecke_matrix(self, n):
r"""
EXAMPLES::
sage: CuspForms(GammaH(31, [7]), 1).hecke_matrix(7)
[-1]
sage: C = CuspForms(GammaH(124, [33]), 1) # long time
sage: C.hecke_matrix(2) # long time
[ 0 0 -1 -1 0 1 0]
[ 1 0 0 -1 -1 -1 0]
[ 0 0 0 -1 1 1 -1]
[ 0 1 0 -1 0 0 0]
[ 0 0 -1 0 0 1 1]
[ 0 0 0 -1 0 0 -1]
[ 0 0 0 0 0 1 0]
sage: C.hecke_matrix(7) # long time
[ 0 1 0 -1 0 0 1]
[ 0 -1 0 0 0 0 0]
[ 0 1 -1 0 0 0 1]
[ 0 0 0 -1 0 0 0]
[ 0 1 1 0 0 -1 0]
[ 1 0 -1 -1 -1 0 1]
[ 0 1 0 0 1 0 0]
sage: C.hecke_matrix(23) == 0 # long time
True
"""
chars = self.group().characters_mod_H(sign=-1, galois_orbits=True)
A = Matrix(QQ, 0, 0)
for c in chars:
chi = c.minimize_base_ring()
d = weight1.dimension_wt1_cusp_forms(chi)
e = chi.base_ring().degree()
H = Matrix(QQ, d * e, d * e)
from .constructor import CuspForms
M = CuspForms(chi, 1).hecke_matrix(n)
if e == 1:
H = M
else:
for i in range(d):
for j in range(d):
H[e * i:e * (i + 1),
e * j:e * (j + 1)] = M[i, j].matrix().transpose()
A = A.block_sum(H)
t = self._transformation_matrix()
return t * A * ~t
def _compute_diamond_matrix(self, d):
r"""
EXAMPLES::
sage: CuspForms(GammaH(31, [7]), 1).diamond_bracket_matrix(3)
[-1]
sage: C = CuspForms(GammaH(124, [33]), 1) # long time
sage: D = C.diamond_bracket_matrix(3); D # long time
[ 0 0 0 -1 1 0 0]
[ 0 -1 0 0 0 0 0]
[ 2 1 1 -2 -1 -2 -1]
[ 0 0 0 -1 0 0 0]
[-1 0 0 1 1 0 0]
[ 2 0 0 -2 -1 -1 0]
[ 0 2 1 0 0 -1 0]
sage: t = C._transformation_matrix(); ~t * D * t # long time
[ 1 1 0 0 0 0 0]
[-1 0 0 0 0 0 0]
[ 0 0 1 1 0 0 0]
[ 0 0 -1 0 0 0 0]
[ 0 0 0 0 -1 0 0]
[ 0 0 0 0 0 -1 0]
[ 0 0 0 0 0 0 -1]
"""
chars=self.group().characters_mod_H(sign=-1, galois_orbits=True)
A = Matrix(QQ, 0, 0)
for c in chars:
chi = c.minimize_base_ring()
dim = weight1.dimension_wt1_cusp_forms(chi)
if chi.base_ring() == QQ:
m = Matrix(QQ, 1, 1, [chi(d)])
else:
m = chi(d).matrix().transpose()
for i in range(dim):
A = A.block_sum(m)
t = self._transformation_matrix()
return t * A * ~t