def space(self):
r'''
Calculates the homology space as a Z-module.
'''
verb = get_verbose()
set_verbose(0)
V = self.coefficient_module()
R = V.base_ring()
Vdim = V.dimension()
G = self.group()
gens = G.gens()
ambient = R**(Vdim * len(gens))
if self.trivial_action():
cycles = ambient
else:
# Now find the subspace of cycles
A = Matrix(R, Vdim, 0)
for g in gens:
for v in V.gens():
A = A.augment(matrix(R,Vdim,1,list(vector(g**-1 * v - v))))
K = A.right_kernel_matrix()
cycles = ambient.submodule([ambient(list(o)) for o in K.rows()])
boundaries = []
for r in G.get_relation_words():
grad = self.twisted_fox_gradient(G(r).word_rep)
for v in V.gens():
boundaries.append(cycles(ambient(sum([list(a * vector(v)) for a in grad],[]))))
boundaries = cycles.submodule(boundaries)
ans = cycles.quotient(boundaries)
set_verbose(verb)
return ans
def _compute_q_expansion_basis(self, prec=None):
r"""
Compute q-expansion basis using Schaeffer's algorithm.
EXAMPLES::
sage: CuspForms(GammaH(31, [7]), 1).q_expansion_basis() # indirect doctest
[
q - q^2 - q^5 + O(q^6)
]
A more elaborate example (two Galois-conjugate characters each giving a
2-dimensional space)::
sage: CuspForms(GammaH(124, [85]), 1).q_expansion_basis()
[
q - q^4 - q^6 + O(q^7),
q^2 + O(q^7),
q^3 + O(q^7),
q^5 - q^6 + O(q^7)
]
"""
if prec is None:
prec = self.prec()
else:
prec = Integer(prec)
chars=self.group().characters_mod_H(sign=-1, galois_orbits=True)
B = []
dim = 0
for c in chars:
chi = c.minimize_base_ring()
Bchi = [weight1.modular_ratio_to_prec(chi, f, prec)
for f in weight1.hecke_stable_subspace(chi) ]
if Bchi == []:
continue
if chi.base_ring() == QQ:
B += [f.padded_list(prec) for f in Bchi]
dim += len(Bchi)
else:
d = chi.base_ring().degree()
dim += d * len(Bchi)
for f in Bchi:
w = f.padded_list(prec)
for i in range(d):
B.append([x[i] for x in w])
basis_mat = Matrix(QQ, B)
if basis_mat.is_zero():
return []
# Daft thing: "transformation=True" parameter to echelonize
# is ignored for rational matrices!
big_mat = basis_mat.augment(identity_matrix(dim))
big_mat.echelonize()
c = big_mat.pivots()[-1]
echelon_basis_mat = big_mat[:, :prec]
R = self._q_expansion_ring()
if c >= prec:
verbose("Precision %s insufficient to determine basis" % prec, level=1)
else:
verbose("Minimal precision for basis: %s" % (c+1), level=1)
t = big_mat[:, prec:]
assert echelon_basis_mat == t * basis_mat
self.__transformation_matrix = t
self._char_basis = [R(f.list(), c+1) for f in basis_mat.rows()]
return [R(f.list(), prec) for f in echelon_basis_mat.rows() if f != 0]
