2537.sagews

Function to calculate classical Kloosterman sum $S(m,n,c)=\sum_{\substack{
   d\text{(mod c)} \\
   gcd(c,d)=1
  }}
 e\left(\frac{m\overline{d}+nd}{c}\right)$

def klsum(m,n,c):
    sum=1
    for d in range(1,c-1):
        if gcd(d,c) == 1:
            dinv=inverse_mod(d,c)
            z=m*dinv+n*d
            sum=sum+exp(2*pi * i * z / c)
    return sum

Example $S(2,1,5)$

CC(klsum(2,1,5))

-1.42705098312484 - 0.587785252292472*I