def get_satake_parameters(k, N=1, chi=0, fi=0, prec=10, bits=53, angles=False):
r""" Compute the Satake parameters and return an html-table.
INPUT:
- ''k'' -- positive integer : the weight
- ''N'' -- positive integer (default 1) : level
- ''chi'' -- non-neg. integer (default 0) use character nr. chi - ''fi'' -- non-neg. integer (default 0) We want to use the element nr. fi f=Newforms(N,k)[fi]
-''prec'' -- compute parameters for p <=prec
-''bits'' -- do real embedings intoi field of bits precision
-''angles''-- return the angles t_p instead of the alpha_p
here alpha_p=p^((k-1)/2)exp(i*t_p)
"""
(t, f) = _get_newform(k, N, chi, fi)
if(not t):
return f
K = f.base_ring()
RF = RealField(bits)
CF = ComplexField(bits)
if(K != QQ):
M = len(K.complex_embeddings())
ems = dict()
for j in range(M):
ems[j] = list()
ps = prime_range(prec)
alphas = list()
for p in ps:
ap = f.coefficients(ZZ(prec))[p]
if(K == QQ):
f1 = QQ(4 * p ** (k - 1) - ap ** 2)
alpha_p = (QQ(ap) + I * f1.sqrt()) / QQ(2)
# beta_p=(QQ(ap)-I*f1.sqrt())/QQ(2)
# satake[p]=(alpha_p,beta_p)
ab = RF(p ** ((k - 1) / 2))
norm_alpha = alpha_p / ab
t_p = CF(norm_alpha).argument()
if(angles):
alphas.append(t_p)
else:
alphas.append(alpha_p)
else:
for j in range(M):
app = ap.complex_embeddings(bits)[j]
f1 = (4 * p ** (k - 1) - app ** 2)
alpha_p = (app + I * f1.sqrt()) / RealField(bits)(2)
ab = RF(p ** ((k - 1) / 2))
norm_alpha = alpha_p / ab
t_p = CF(norm_alpha).argument()
if(angles):
ems[j].append(t_p)
else:
ems[j].append(alpha_p)
tbl = dict()
tbl['headersh'] = ps
if(K == QQ):
tbl['headersv'] = [""]
tbl['data'] = [alphas]
tbl['corner_label'] = "$p$"
else:
tbl['data'] = list()
tbl['headersv'] = list()
tbl['corner_label'] = "Embedding \ $p$"
for j in ems.keys():
tbl['headersv'].append(j)
tbl['data'].append(ems[j])
# logger.debug(tbl)
s = html_table(tbl)
return s
def get_satake_parameters(k, N=1, chi=0, fi=0, prec=10, bits=53, angles=False):
r""" Compute the Satake parameters and return an html-table.
INPUT:
- ''k'' -- positive integer : the weight
- ''N'' -- positive integer (default 1) : level
- ''chi'' -- non-neg. integer (default 0) use character nr. chi - ''fi'' -- non-neg. integer (default 0) We want to use the element nr. fi f=Newforms(N,k)[fi]
-''prec'' -- compute parameters for p <=prec
-''bits'' -- do real embedings intoi field of bits precision
-''angles''-- return the angles t_p instead of the alpha_p
here alpha_p=p^((k-1)/2)exp(i*t_p)
"""
(t, f) = _get_newform(k, N, chi, fi)
if (not t):
return f
K = f.base_ring()
RF = RealField(bits)
CF = ComplexField(bits)
if (K != QQ):
M = len(K.complex_embeddings())
ems = dict()
for j in range(M):
ems[j] = list()
ps = prime_range(prec)
alphas = list()
for p in ps:
ap = f.coefficients(ZZ(prec))[p]
if (K == QQ):
f1 = QQ(4 * p**(k - 1) - ap**2)
alpha_p = (QQ(ap) + I * f1.sqrt()) / QQ(2)
# beta_p=(QQ(ap)-I*f1.sqrt())/QQ(2)
# satake[p]=(alpha_p,beta_p)
ab = RF(p**((k - 1) / 2))
norm_alpha = alpha_p / ab
t_p = CF(norm_alpha).argument()
if (angles):
alphas.append(t_p)
else:
alphas.append(alpha_p)
else:
for j in range(M):
app = ap.complex_embeddings(bits)[j]
f1 = (4 * p**(k - 1) - app**2)
alpha_p = (app + I * f1.sqrt()) / RealField(bits)(2)
ab = RF(p**((k - 1) / 2))
norm_alpha = alpha_p / ab
t_p = CF(norm_alpha).argument()
if (angles):
ems[j].append(t_p)
else:
ems[j].append(alpha_p)
tbl = dict()
tbl['headersh'] = ps
if (K == QQ):
tbl['headersv'] = [""]
tbl['data'] = [alphas]
tbl['corner_label'] = "$p$"
else:
tbl['data'] = list()
tbl['headersv'] = list()
tbl['corner_label'] = "Embedding \ $p$"
for j in ems.keys():
tbl['headersv'].append(j)
tbl['data'].append(ems[j])
# logger.debug(tbl)
s = html_table(tbl)
return s
def compute_satake_parameters_numeric(self, prec=10, bits=53,insert_in_db=True):
r""" Compute the Satake parameters and return an html-table.
We only do satake parameters for primes p primitive to the level.
By defintion the S. parameters are given as the roots of
X^2 - c(p)X + chi(p)*p^(k-1) if (p,N)=1
INPUT:
-''prec'' -- compute parameters for p <=prec
-''bits'' -- do real embedings intoi field of bits precision
"""
if self.character().order()>2:
## We only implement this for trival or quadratic characters.
## Otherwise there is difficulty to figure out what the embeddings mean...
return
K = self.coefficient_field()
degree = self.degree()
RF = RealField(bits)
CF = ComplexField(bits)
ps = prime_range(prec)
self._satake['ps'] = []
alphas = dict()
thetas = dict()
aps = list()
tps = list()
k = self.weight()
for j in range(degree):
alphas[j] = dict()
thetas[j] = dict()
for j in xrange(len(ps)):
p = ps[j]
try:
ap = self.coefficient(p)
except IndexError:
break
# Remove bad primes
if p.divides(self.level()):
continue
self._satake['ps'].append(p)
chip = self.character().value(p)
wmf_logger.debug("p={0}".format(p))
wmf_logger.debug("chip={0} of type={1}".format(chip,type(chip)))
if hasattr(chip,'complex_embeddings'):
wmf_logger.debug("embeddings(chip)={0}".format(chip.complex_embeddings()))
wmf_logger.debug("ap={0}".format(ap))
wmf_logger.debug("K={0}".format(K))
# ap=self._f.coefficients(ZZ(prec))[p]
if K.absolute_degree()==1:
f1 = QQ(4 * chip * p ** (k - 1) - ap ** 2)
alpha_p = (QQ(ap) + I * f1.sqrt()) / QQ(2)
ab = RF(p ** ((k - 1) / 2))
norm_alpha = alpha_p / ab
t_p = CF(norm_alpha).argument()
thetas[0][p] = t_p
alphas[0][p] = (alpha_p / ab).n(bits)
else:
for jj in range(degree):
app = ap.complex_embeddings(bits)[jj]
wmf_logger.debug("chip={0}".format(chip))
wmf_logger.debug("app={0}".format(app))
wmf_logger.debug("jj={0}".format(jj))
if not hasattr(chip,'complex_embeddings'):
f1 = (4 * CF(chip) * p ** (k - 1) - app ** 2)
else:
f1 = (4 * chip.complex_embeddings(bits)[jj] * p ** (k - 1) - app ** 2)
alpha_p = (app + I * abs(f1).sqrt())
# ab=RF(/RF(2)))
# alpha_p=alpha_p/RealField(bits)(2)
wmf_logger.debug("f1={0}".format(f1))
alpha_p = alpha_p / RF(2)
wmf_logger.debug("alpha_p={0}".format(alpha_p))
t_p = CF(alpha_p).argument()
# tps.append(t_p)
# aps.append(alpha_p)
alphas[jj][p] = alpha_p
thetas[jj][p] = t_p
self._satake['alphas'] = alphas
self._satake['thetas'] = thetas
self._satake['alphas_latex'] = dict()
self._satake['thetas_latex'] = dict()
for j in self._satake['alphas'].keys():
self._satake['alphas_latex'][j] = dict()
for p in self._satake['alphas'][j].keys():
s = latex(self._satake['alphas'][j][p])
self._satake['alphas_latex'][j][p] = s
for j in self._satake['thetas'].keys():
self._satake['thetas_latex'][j] = dict()
for p in self._satake['thetas'][j].keys():
s = latex(self._satake['thetas'][j][p])
self._satake['thetas_latex'][j][p] = s
wmf_logger.debug("satake=".format(self._satake))
return self._satake