 Gauss sum of a Dirichlet character

Extend a Dirichlet character by zero on nonunits. Its Gauss sum is $g(\chi )={\sum }_{a\mathrm{mod}N}\chi (a)e^{2\pi ia/N}$, and its finite Fourier transform at $r$ is $S_{\chi }(r)={\sum }_{a\mathrm{mod}N}\chi (a)e^{2\pi ira/N}$.