def kloosterman_sum(self, arg):
a, b = map(int, arg.split(','))
modulus, number = self.modulus, self.number
if modulus == 1:
# there is a bug in sage for modulus = 1
return r"""
\( \displaystyle K(%s,%s,\chi_{1}(1,·))
= \sum_{r \in \Z/\Z}
\chi_{1}(1,r) 1^{%s r + %s r^{-1}}
= 1 \)
""" % (a, b, a, b)
chi_values_data = db.char_dir_values.lookup(
"{}.{}".format(modulus, number)
)
chi_valuepairs = chi_values_data['values_gens']
chi_genvalues = [int(v) for g, v in chi_valuepairs]
chi = self.chi.sage_character(self.order, chi_genvalues)
k = chi.kloosterman_sum_numerical(100, a, b)
k = complex2str(k, 10)
return r"""
\( \displaystyle K(%s,%s,\chi_{%s}(%s,·))
= \sum_{r \in \Z/%s\Z}
\chi_{%s}(%s,r) e\left(\frac{%s r + %s r^{-1}}{%s}\right)
= %s \)""" % (a, b, modulus, number, modulus, modulus, number, a, b, modulus, k)
def gauss_sum(self, val):
val = int(val)
mod, num = self.modulus, self.number
g = self.chi.gauss_sum_numerical(val)
g = complex2str(g)
x = Rational('%s/%s' % (val, mod))
n = x.numerator()
n = str(n) + "r" if not n == 1 else "r"
d = x.denominator()
Gtex = r'\Z/%s\Z' % mod
chitex = self.char2tex(mod, num, tag=False)
chitexr = self.char2tex(mod, num, 'r', tag=False)
deftex = r'\sum_{r\in %s} %s e\left(\frac{%s}{%s}\right)'%(Gtex,chitexr,n,d)
return r"\(\displaystyle \tau_{%s}(%s) = %s = %s \)" % (val, chitex, deftex, g)
def gauss_sum(self, val):
val = int(val)
mod, num = self.modulus, self.number
chi = self.chi.sage_character()
g = chi.gauss_sum_numerical(100, val)
g = complex2str(g)
from sage.rings.rational import Rational
x = Rational('%s/%s' % (val, mod))
n = x.numerator()
n = str(n) + "r" if not n == 1 else "r"
d = x.denominator()
Gtex = '\Z/%s\Z' % mod
chitex = self.char2tex(mod, num, tag=False)
chitexr = self.char2tex(mod, num, 'r', tag=False)
deftex = r'\sum_{r\in %s} %s e\left(\frac{%s}{%s}\right)'%(Gtex,chitexr,n,d)
return r"\(\displaystyle \tau_{%s}(%s) = %s = %s. \)" % (val, chitex, deftex, g)
def kloosterman_sum(self, arg):
a, b = map(int, arg.split(','))
modulus, number = self.modulus, self.number
if modulus == 1:
# there is a bug in sage for modulus = 1
return r"""
\( \displaystyle K(%s,%s,\chi_{1}(1,·))
= \sum_{r \in \Z/\Z}
\chi_{1}(1,r) 1^{%s r + %s r^{-1}}
= 1 \)
""" % (a, b, a, b)
chi = self.chi.sage_character()
k = chi.kloosterman_sum_numerical(100, a, b)
k = complex2str(k, 10)
return r"""
\( \displaystyle K(%s,%s,\chi_{%s}(%s,·))
= \sum_{r \in \Z/%s\Z}
\chi_{%s}(%s,r) e\left(\frac{%s r + %s r^{-1}}{%s}\right)
= %s. \)""" % (a, b, modulus, number, modulus, modulus, number, a, b, modulus, k)