def value(self, z, embedding=0):
if self.prec == 0:
return 0
else:
q = exp(2 * CC.pi() * CC(0, 1) * z)
return sum(
self.coefficient_embedding(n, embedding) * q**n
for n in range(self.prec))
def check_ap2_slow(rec):
# Check a_{p^2} = a_p^2 - chi(p) for primes up to 31
ls = rec['lfunction_label'].split('.')
level, weight, chi = map(int, [ls[0], ls[1], ls[-2]])
char = DirichletGroup_conrey(level, CC)[chi]
Z = rec['an_normalized[0:1000]']
for p in prime_range(31+1):
if level % p != 0:
# a_{p^2} = a_p^2 - chi(p)
charval = CC(2*char.logvalue(int(p)) * CC.pi()*CC.gens()[0]).exp()
else:
charval = 0
if (CC(*Z[p**2 - 1]) - (CC(*Z[p-1])**2 - charval)).abs() > 1e-11:
return False
return True
def check_ap2_slow(rec):
# Check a_{p^2} = a_p^2 - chi(p) for primes up to 31
ls = rec['lfunction_label'].split('.')
level, weight, chi = map(int, [ls[0], ls[1], ls[-2]])
char = DirichletGroup_conrey(level, CC)[chi]
Z = rec['an_normalized[0:1000]']
for p in prime_range(31+1):
if level % p != 0:
# a_{p^2} = a_p^2 - chi(p)
charval = CC(2*char.logvalue(int(p)) * CC.pi()*CC.gens()[0]).exp()
else:
charval = 0
if (CC(*Z[p**2 - 1]) - (CC(*Z[p-1])**2 - charval)).abs() > 1e-11:
return False
return True
def check_ap2_slow(self, rec, verbose=False):
"""
Check a_{p^2} = a_p^2 - chi(p) for primes up to 31
"""
ls = rec['label'].split('.')
level, weight, chi = map(int, [ls[0], ls[1], ls[-2]])
char = DirichletGroup_conrey(level, CC)[chi]
Z = rec['an_normalized']
for p in prime_range(31+1):
if level % p != 0:
# a_{p^2} = a_p^2 - chi(p)
charval = CC(2*char.logvalue(int(p)) * CC.pi()*CC.gens()[0]).exp()
else:
charval = 0
if (CC(*Z[p**2 - 1]) - (CC(*Z[p-1])**2 - charval)).abs() > 1e-13:
if verbose:
print "ap2 failure", p, CC(*Z[p**2 - 1]), CC(*Z[p-1])**2 - charval
return False
return True
def value(self, z, embedding=0):
if self.prec == 0:
return 0
else:
q = exp(2*CC.pi()*CC(0,1)*z)
return sum(self.coefficient_embedding(n,embedding)*q**n for n in range(self.prec))
def gaussum(n, N, prec=53):
CC = ComplexField(prec)
return sum(CC(exp(2 * CC.pi() * CC(0, 1) * n * m ** 2 / N)) for m in range(N))
def gaussum(n, N, prec=53):
CC = ComplexField(prec)
return sum(
CC(exp(2 * CC.pi() * CC(0, 1) * n * m**2 / N)) for m in range(N))