def get_values_at_CM_points(k, N=1, chi=0, fi=0, digits=12, verbose=0):
r""" Computes and returns a list of values of f at a collection of CM points as complex floating point numbers.
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]
-''digits'' -- we want this number of corrrect digits in the value
OUTPUT:
-''s'' string representation of a dictionary {I:f(I):rho:f(rho)}.
TODO: Get explicit, algebraic values if possible!
"""
(t, f) = _get_newform(k, N, chi, fi)
if(not t):
return f
bits = max(53, ceil(digits * 4))
CF = ComplexField(bits)
RF = ComplexField(bits)
eps = RF(10 ** -(digits + 1))
logger.debug("eps=" % eps)
K = f.base_ring()
cm_vals = dict()
# the points we want are i and rho. More can be added later...
rho = CyclotomicField(3).gen()
zi = CyclotomicField(4).gen()
points = [rho, zi]
maxprec = 1000 # max size of q-expansion
minprec = 10 # max size of q-expansion
for tau in points:
q = CF(exp(2 * pi * I * tau))
fexp = dict()
if(K == QQ):
v1 = CF(0)
v2 = CF(1)
try:
for prec in range(minprec, maxprec, 10):
logger.debug("prec=%s" % prec)
v2 = f.q_expansion(prec)(q)
err = abs(v2 - v1)
logger.debug("err=%s" % err)
if(err < eps):
raise StopIteration()
v1 = v2
cm_vals[tau] = ""
except StopIteration:
cm_vals[tau] = str(fq)
else:
v1 = dict()
v2 = dict()
err = dict()
cm_vals[tau] = dict()
for h in range(K.degree()):
v1[h] = 1
v2[h] = 0
try:
for prec in range(minprec, maxprec, 10):
logger.debug("prec=%s" % prec)
for h in range(K.degree()):
fexp[h] = list()
v2[h] = 0
for n in range(prec):
c = f.coefficients(ZZ(prec))[n]
cc = c.complex_embeddings(CF.prec())[h]
v2[h] = v2[h] + cc * q ** n
err[h] = abs(v2[h] - v1[h])
logger.debug("v1[%s]=%s" % (h, v1[h]))
logger.debug("v2[%s]=%s" % (h, v2[h]))
logger.debug("err[%s]=%s" % (h, err[h]))
if(max(err.values()) < eps):
raise StopIteration()
v1[h] = v2[h]
except StopIteration:
pass
for h in range(K.degree()):
if(err[h] < eps):
cm_vals[tau][h] = v2[h]
else:
cm_vals[tau][h] = ""
logger.debug("vals=%s" % cm_vals)
logger.debug("errs=%s" % err)
tbl = dict()
tbl['corner_label'] = ['$\tau$']
tbl['headersh'] = ['$\\rho=\zeta_{3}$', '$i$']
if(K == QQ):
tbl['headersv'] = ['$f(\\tau)$']
tbl['data'] = [cm_vals]
else:
tbl['data'] = list()
tbl['headersv'] = list()
for h in range(K.degree()):
tbl['headersv'].append("$\sigma_{%s}(f(\\tau))$" % h)
row = list()
for tau in points:
row.append(cm_vals[tau][h])
tbl['data'].append(row)
s = html_table(tbl)
# s=html.table([cm_vals.keys(),cm_vals.values()])
return s
def get_values_at_CM_points(k, N=1, chi=0, fi=0, digits=12, verbose=0):
r""" Computes and returns a list of values of f at a collection of CM points as complex floating point numbers.
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]
-''digits'' -- we want this number of corrrect digits in the value
OUTPUT:
-''s'' string representation of a dictionary {I:f(I):rho:f(rho)}.
TODO: Get explicit, algebraic values if possible!
"""
(t, f) = _get_newform(k, N, chi, fi)
if (not t):
return f
bits = max(53, ceil(digits * 4))
CF = ComplexField(bits)
RF = ComplexField(bits)
eps = RF(10**-(digits + 1))
logger.debug("eps=" % eps)
K = f.base_ring()
cm_vals = dict()
# the points we want are i and rho. More can be added later...
rho = CyclotomicField(3).gen()
zi = CyclotomicField(4).gen()
points = [rho, zi]
maxprec = 1000 # max size of q-expansion
minprec = 10 # max size of q-expansion
for tau in points:
q = CF(exp(2 * pi * I * tau))
fexp = dict()
if (K == QQ):
v1 = CF(0)
v2 = CF(1)
try:
for prec in range(minprec, maxprec, 10):
logger.debug("prec=%s" % prec)
v2 = f.q_expansion(prec)(q)
err = abs(v2 - v1)
logger.debug("err=%s" % err)
if (err < eps):
raise StopIteration()
v1 = v2
cm_vals[tau] = ""
except StopIteration:
cm_vals[tau] = str(fq)
else:
v1 = dict()
v2 = dict()
err = dict()
cm_vals[tau] = dict()
for h in range(K.degree()):
v1[h] = 1
v2[h] = 0
try:
for prec in range(minprec, maxprec, 10):
logger.debug("prec=%s" % prec)
for h in range(K.degree()):
fexp[h] = list()
v2[h] = 0
for n in range(prec):
c = f.coefficients(ZZ(prec))[n]
cc = c.complex_embeddings(CF.prec())[h]
v2[h] = v2[h] + cc * q**n
err[h] = abs(v2[h] - v1[h])
logger.debug("v1[%s]=%s" % (h, v1[h]))
logger.debug("v2[%s]=%s" % (h, v2[h]))
logger.debug("err[%s]=%s" % (h, err[h]))
if (max(err.values()) < eps):
raise StopIteration()
v1[h] = v2[h]
except StopIteration:
pass
for h in range(K.degree()):
if (err[h] < eps):
cm_vals[tau][h] = v2[h]
else:
cm_vals[tau][h] = ""
logger.debug("vals=%s" % cm_vals)
logger.debug("errs=%s" % err)
tbl = dict()
tbl['corner_label'] = ['$\tau$']
tbl['headersh'] = ['$\\rho=\zeta_{3}$', '$i$']
if (K == QQ):
tbl['headersv'] = ['$f(\\tau)$']
tbl['data'] = [cm_vals]
else:
tbl['data'] = list()
tbl['headersv'] = list()
for h in range(K.degree()):
tbl['headersv'].append("$\sigma_{%s}(f(\\tau))$" % h)
row = list()
for tau in points:
row.append(cm_vals[tau][h])
tbl['data'].append(row)
s = html_table(tbl)
# s=html.table([cm_vals.keys(),cm_vals.values()])
return s
def compute_cm_values_numeric(self,digits=12,insert_in_db=True):
r"""
Compute CM-values numerically.
"""
if isinstance(self._cm_values,dict) and self._cm_values <> {}:
return self._cm_values
# the points we want are i and rho. More can be added later...
bits = ceil(int(digits) * int(4))
CF = ComplexField(bits)
RF = ComplexField(bits)
eps = RF(10 ** - (digits + 1))
if(self._verbose > 1):
wmf_logger.debug("eps={0}".format(eps))
K = self.base_ring()
# recall that
degree = self.degree()
cm_vals = dict()
rho = CyclotomicField(3).gen()
zi = CyclotomicField(4).gen()
points = [rho, zi]
maxprec = 1000 # max size of q-expansion
minprec = 10 # max size of q-expansion
for tau in points:
q = CF(exp(2 * pi * I * tau))
fexp = dict()
cm_vals[tau] = dict()
if(tau == I and self.level() == -1):
# cv= #"Exact(soon...)" #_cohen_exact_formula(k)
for h in range(degree):
cm_vals[tau][h] = cv
continue
if K.absolute_degree()==1:
v1 = CF(0)
v2 = CF(1)
try:
for prec in range(minprec, maxprec, 10):
if(self._verbose > 1):
wmf_logger.debug("prec={0}".format(prec))
v2 = self.as_factor().q_eigenform(prec).truncate(prec)(q)
err = abs(v2 - v1)
if(self._verbose > 1):
wmf_logger.debug("err={0}".format(err))
if(err < eps):
raise StopIteration()
v1 = v2
cm_vals[tau][0] = None
except StopIteration:
cm_vals[tau][0] = v2
else:
v1 = dict()
v2 = dict()
err = dict()
for h in range(degree):
v1[h] = 1
v2[h] = 0
try:
for prec in range(minprec, maxprec, 10):
if(self._verbose > 1):
wmf_logger.debug("prec={0}".format(prec))
c = self.coefficients(range(prec),insert_in_db=insert_in_db)
for h in range(degree):
fexp[h] = list()
v2[h] = 0
for n in range(prec):
cn = c[n]
if hasattr(cn, 'complex_embeddings'):
cc = cn.complex_embeddings(CF.prec())[h]
else:
cc = CF(cn)
v2[h] = v2[h] + cc * q ** n
err[h] = abs(v2[h] - v1[h])
if(self._verbose > 1):
wmf_logger.debug("v1[{0}]={1}".format(h,v1[h]))
wmf_logger.debug("v2[{0}]={1}".format(h,v2[h]))
wmf_logger.debug("err[{0}]={2}".format(h,err[h]))
if(max(err.values()) < eps):
raise StopIteration()
v1[h] = v2[h]
except StopIteration:
pass
for h in range(degree):
if(err[h] < eps):
cm_vals[tau][h] = v2[h]
else:
cm_vals[tau][h] = None
self._cm_values = cm_vals
