There is a wide litterature for the classical Gauss sums. For $\chi$ a primitive Dirichlet character modulo $N$, it is given by $$\tau(\chi) = \sum_{n \text{ mod } N} \chi(n) \exp(2i\pi n/N).$$
An interesting fact is that primitive Dirichlet characters are essentially their own Fourier transform, up to this associated Gauss sum: $$\sum_{n \text{ mod } N} \chi(n) \exp(2i\pi nm/N) = \bar{\chi} (m) \tau(\chi). \qquad (\star)$$
I am interested in similar properties in general number fields $F$. Let $\omega$ be a nonzero element of $F$ and write $\mathfrak{a}$ for its denominator. Hecke defined some analogous Gauss sums by $$C(\omega) = \sum_{n \text{ mod } \mathfrak{a}} \exp(\mathrm{tr}(n^2 \omega)).$$
(here $n \text{ mod } \mathfrak{a}$ means that $n$ is an integer ideal in $\mathfrak{o}_F/\mathfrak{a}\mathfrak{o}_F$, where $\mathfrak{o}_F$ is the ring of integers of $F$). I have not found anything concerning the analogous Gauss sums as above, namely for a finite order character $\chi$ on integer ideals which is a character modulo $\mathfrak{a}$, $$\tau(\omega, \chi) = \sum_{n \text{ mod } \mathfrak{a}} \chi(n)\exp(\mathrm{tr}(n^2 \omega)).$$
Is there anything of this kind in the litterature? In particular, do we have an analogous result to $(\star)$?
