Gauss sum of a Dirichlet character ($g(\chi )={\sum }_{a\mathrm{mod}N}\chi (a)e^{2\pi ia/N}$)

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 $S_{\chi }(r)={\sum }_{a\mathrm{mod}N}\chi (a)e^{2\pi ira/N}$.

For a primitive Dirichlet character, $S_{\chi }(r)=\overline{\chi }(r)g(\chi )$ for every integer $r$, with zero on nonunits. For a unit $r$ this follows by substitution. If a prime $p$ divides both $r$ and $N$, primitivity supplies a unit $u\equiv 1(\mathrm{mod}N/p)$ with $\chi (u)\ne 1$; substitution by $u$ fixes the additive exponential and forces $S_{\chi }(r)=0$. For an imprimitive Dirichlet character this vanishing can fail.