TITLE: Gauss sums for general number fields
QUESTION [9 upvotes]: 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 $(\star)$?

REPLY [3 votes]: Hecke Gauss sums are quadratic Gauss sums, and (*) is correct for quadratic characters (Hecke, Satz 155). Hecke Gauss sums were studied later by

Siegel, Über das quadratische Reziprozitätsgesetz in algebraischen
Zahlkörpern, Nachr. Akad. Wiss. Göttingen, Math. Phys. Kl. II
(1960), 1-16; Ges. Abh. 3 (1966), 334-349
Shiratani, On the decomposition laws of rational primes in certain
class 2 extensions, Investigations in number theory, 
Advances in Pure Math. 13 (1988), 345-411 
Shiratani, On the Gauss-Hecke-sums, J. Math. Soc. Japan 16 (1964), 32-38
Boylan, Skoruppa, A quick proof of reciprocity for Hecke Gauss sums,
J. Number Theory 133 (2013), 110-114.