def __find_eisen_chars_gammaH(N, H, k):
"""
Find all triples `(\psi_1, \psi_2, t)` that give rise to an Eisenstein series of weight `k` on
`\Gamma_H(N)`.
EXAMPLES::
sage: pars = sage.modular.modform.eis_series.__find_eisen_chars_gammaH(15, [2], 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), 1), ((1, -1), (-1, 1), 1), ((-1, -1), (1, 1), 1)]
"""
params = []
for chi in dirichlet.DirichletGroup(N):
if all([chi(h) == 1 for h in H]):
params += __find_eisen_chars(chi, k)
return params
def _modular_symbols_space_character(self):
"""
Return a random space of modular symbols for Gamma1, with
(possibly trivial) character, random sign, and weight a
random choice from self.weights.
EXAMPLES:
sage: sage.modular.modsym.tests.Test()._modular_symbols_space_character() # random
level = 18, weight = 3, sign = 0
Modular Symbols space of dimension 0 and level 18, weight 3, character [1, zeta6 - 1], sign 0, over Cyclotomic Field of order 6 and degree 2
"""
level, weight, sign = self._level_weight_sign()
G = dirichlet.DirichletGroup(level)
eps = G.random_element()
M = modsym.ModularSymbols(eps, weight, sign)
self.current_space = M
return M
def __find_eisen_chars_gamma1(N, k):
"""
Find all triples `(\psi_1, \psi_2, t)` that give rise to an Eisenstein series of weight `k` on
`\Gamma_1(N)`.
EXAMPLES::
sage: pars = sage.modular.modform.eis_series.__find_eisen_chars_gamma1(12, 4)
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), 2),
((1, 1), (1, 1), 3),
((1, 1), (1, 1), 4),
((1, 1), (1, 1), 6),
((1, 1), (1, 1), 12),
((1, 1), (-1, -1), 1),
((-1, -1), (1, 1), 1),
((-1, 1), (1, -1), 1),
((1, -1), (-1, 1), 1)]
sage: pars = sage.modular.modform.eis_series.__find_eisen_chars_gamma1(12, 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), 1),
((-1, 1), (1, 1), 3),
((1, 1), (1, -1), 1),
((1, 1), (1, -1), 2),
((1, 1), (1, -1), 4),
((1, -1), (1, 1), 1),
((1, -1), (1, 1), 2),
((1, -1), (1, 1), 4)]
"""
pairs = []
s = (-1)**k
G = dirichlet.DirichletGroup(N)
E = list(G)
parity = [c(-1) for c in E]
for i in range(len(E)):
for j in range(i,len(E)):
if parity[i]*parity[j] == s and N % (E[i].conductor()*E[j].conductor()) == 0:
chi, psi = __common_minimal_basering(E[i], E[j])
if k != 1:
pairs.append((chi, psi))
if i!=j: pairs.append((psi,chi))
else:
# if weight is 1 then (chi, psi) and (chi, psi) are the
# same form
if psi.is_trivial() and not chi.is_trivial():
# need to put the trivial character first to get the L-value right
pairs.append((psi, chi))
else:
pairs.append((chi, psi))
#end fors
#end if
triples = []
D = divisors(N)
for chi, psi in pairs:
c_chi = chi.conductor()
c_psi = psi.conductor()
D = divisors(N/(c_chi * c_psi))
if (k==2 and chi.is_trivial() and psi.is_trivial()):
D.remove(1)
chi, psi = __common_minimal_basering(chi, psi)
for t in D:
triples.append((chi, psi, t))
return triples
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 = cputime()
for e in G:
m = Integer(e.conductor())
if m in C:
C[m].append(e)
else:
C[m] = [e]
verbose("Enumeration with conductors.", t0)
params = []
for L in divisors(N):
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
