def __find_eisen_chars(character, k):
"""
Find all triples `(\psi_1, \psi_2, t)` that give rise to an Eisenstein series of the given weight and character.
EXAMPLES::
sage: sage.modular.modform.eis_series.__find_eisen_chars(DirichletGroup(36).0, 4)
[]
sage: pars = sage.modular.modform.eis_series.__find_eisen_chars(DirichletGroup(36).0, 5)
sage: [(x[0].values_on_gens(), x[1].values_on_gens(), x[2]) for x in pars]
[((1, 1), (-1, 1), 1),
((1, 1), (-1, 1), 3),
((1, 1), (-1, 1), 9),
((1, -1), (-1, -1), 1),
((-1, 1), (1, 1), 1),
((-1, 1), (1, 1), 3),
((-1, 1), (1, 1), 9),
((-1, -1), (1, -1), 1)]
"""
N = character.modulus()
if character.is_trivial():
if k%2 != 0:
return []
char_inv = ~character
V = [(character, char_inv, t) for t in divisors(N) if t>1]
if k != 2:
V.insert(0,(character, char_inv, 1))
if is_squarefree(N):
return V
# Now include all pairs (chi,chi^(-1)) such that cond(chi)^2 divides N:
# TODO: Optimize -- this is presumably way too hard work below.
G = dirichlet.DirichletGroup(N)
for chi in G:
if not chi.is_trivial():
f = chi.conductor()
if N % (f**2) == 0:
chi = chi.minimize_base_ring()
chi_inv = ~chi
for t in divisors(N//(f**2)):
V.insert(0, (chi, chi_inv, t))
return V
eps = character
if eps(-1) != (-1)**k:
return []
eps = eps.maximize_base_ring()
G = eps.parent()
# Find all pairs chi, psi such that:
#
# (1) cond(chi)*cond(psi) divides the level, and
#
# (2) chi*psi == eps, where eps is the nebentypus character of self.
#
# See [Miyake, Modular Forms] Lemma 7.1.1.
K = G.base_ring()
C = {}
t0 = misc.cputime()
for e in G:
m = Integer(e.conductor())
if m in C:
C[m].append(e)
else:
C[m] = [e]
misc.verbose("Enumeration with conductors.",t0)
params = []
for L in divisors(N):
misc.verbose("divisor %s"%L)
if L not in C:
continue
GL = C[L]
for R in divisors(N/L):
if R not in C:
continue
GR = C[R]
for chi in GL:
for psi in GR:
if chi*psi == eps:
chi0, psi0 = __common_minimal_basering(chi, psi)
for t in divisors(N//(R*L)):
if k != 1 or ((psi0, chi0, t) not in params):
params.append( (chi0,psi0,t) )
return params
def __find_eisen_chars(character, k):
"""
Find all triples `(\psi_1, \psi_2, t)` that give rise to an Eisenstein series of the given weight and character.
EXAMPLES::
sage: sage.modular.modform.eis_series.__find_eisen_chars(DirichletGroup(36).0, 4)
[]
sage: pars = sage.modular.modform.eis_series.__find_eisen_chars(DirichletGroup(36).0, 5)
sage: [(x[0].values_on_gens(), x[1].values_on_gens(), x[2]) for x in pars]
[((1, 1), (-1, 1), 1),
((1, 1), (-1, 1), 3),
((1, 1), (-1, 1), 9),
((1, -1), (-1, -1), 1),
((-1, 1), (1, 1), 1),
((-1, 1), (1, 1), 3),
((-1, 1), (1, 1), 9),
((-1, -1), (1, -1), 1)]
"""
N = character.modulus()
if character.is_trivial():
if k % 2 != 0:
return []
char_inv = ~character
V = [(character, char_inv, t) for t in divisors(N) if t > 1]
if k != 2:
V.insert(0, (character, char_inv, 1))
if is_squarefree(N):
return V
# Now include all pairs (chi,chi^(-1)) such that cond(chi)^2 divides N:
# TODO: Optimize -- this is presumably way too hard work below.
G = dirichlet.DirichletGroup(N)
for chi in G:
if not chi.is_trivial():
f = chi.conductor()
if N % (f**2) == 0:
chi = chi.minimize_base_ring()
chi_inv = ~chi
for t in divisors(N // (f**2)):
V.insert(0, (chi, chi_inv, t))
return V
eps = character
if eps(-1) != (-1)**k:
return []
eps = eps.maximize_base_ring()
G = eps.parent()
# Find all pairs chi, psi such that:
#
# (1) cond(chi)*cond(psi) divides the level, and
#
# (2) chi*psi == eps, where eps is the nebentypus character of self.
#
# See [Miyake, Modular Forms] Lemma 7.1.1.
K = G.base_ring()
C = {}
t0 = misc.cputime()
for e in G:
m = Integer(e.conductor())
if C.has_key(m):
C[m].append(e)
else:
C[m] = [e]
misc.verbose("Enumeration with conductors.", t0)
params = []
for L in divisors(N):
misc.verbose("divisor %s" % L)
if not C.has_key(L):
continue
GL = C[L]
for R in divisors(N / L):
if not C.has_key(R):
continue
GR = C[R]
for chi in GL:
for psi in GR:
if chi * psi == eps:
chi0, psi0 = __common_minimal_basering(chi, psi)
for t in divisors(N // (R * L)):
if k != 1 or ((psi0, chi0, t) not in params):
params.append((chi0, psi0, t))
return params