Function: znchargauss
Section: number_theoretical
C-Name: znchargauss
Prototype: GGDGb
Help: znchargauss(G, chi, {a=1}): given a Dirichlet character chi on
G = (Z/NZ)^*, return the complex Gauss sum g(chi,a).
Doc: Given a Dirichlet character $\chi$ on $G = (\Z/N\Z)^*$ (see
\kbd{znchar}), return the complex Gauss sum
$$g(\chi,a) = \sum_{n = 1}^N \chi(n) e(a n/N)$$
\bprog
? [G,chi] = znchar(-3); \\ quadratic Gauss sum: I*sqrt(3)
? znchargauss(G,chi)
%2 = 1.7320508075688772935274463415058723670*I
? [G,chi] = znchar(5);
? znchargauss(G,chi) \\ sqrt(5)
%2 = 2.2360679774997896964091736687312762354
? G = znstar(300,1); chi = [1,1,12]~;
? znchargauss(G,chi) / sqrt(300) - exp(2*I*Pi*11/25) \\ = 0
%4 = 2.350988701644575016 E-38 + 1.4693679385278593850 E-39*I
? lfuntheta([G,chi], 1) \\ = 0
%5 = -5.79[...] E-39 - 2.71[...] E-40*I
@eprog