Gauss sums ray class group In Neukirch's book, page 503, in  remark 1 he says that 



He gives a reference for that.  I'm not able to get it.
Could someone at least draw the shape of the indicated Gauss sum, beause I have several unsuccessful attempts.
 A: The citation for the article given by Neukirch is

Allgemeine Theorie der Gaußschen Summen in algebraischen Zahlkörpern. Abh. Deutsch. Akad. Wiss. Berlin. Math.-Nat. Klasse 1 (1951) 4–23

I couldn't find the article itself (not that it would've helped me much since I don't know German), but I do have access to the MathSciNet review, which I will attempt to paraphrase because I am not confident that a direct quote would fall under fair use:
We choose $\chi$, a congruence character of a number field $K$, whose conductor is $\mathfrak{f}\mathfrak{u}$ where $\mathfrak{f}$ is the finite part and $\mathfrak{u}$ is the infinite part, as well as a multiple of the conductor, $\mathfrak{m}\mathfrak{w}$, where similarly, $\mathfrak{m}$ is the finite part and $\mathfrak{w}$ is the infinite part. We choose a divisor $\mathfrak{a}$ of $K$, and define its complement $\widetilde{\mathfrak{a}}$ by the relation $\mathfrak{a}\widetilde{\mathfrak{a}}=\mathfrak{d}^{-1}$ where $\mathfrak{d}$ is the different of $K$. Lastly, we choose $n$, which is a non-zero element of $\widetilde{\mathfrak{a}}\mathfrak{m}^{-1}$.
The Gauss sums considered by Hasse in this article are, apparently, those of the form
$$\tau_n(\chi,\mathfrak{m}\mathfrak{w}|\mathfrak{a})=\textstyle\sum\chi(\mathfrak{x})e(xn).$$
The sum is over the residue class representatives $x$ which satisfy (and here I am quoting, because I don't understand what it means):
$$x\bmod \mathfrak{m}\mathfrak{a},\;x\in\mathfrak{a}; \;x=\mathfrak{x}\mathfrak{a}, \;(\mathfrak{x},\mathfrak{m})=1\,\text{ and }\,x\equiv 1\,\mathord{(}\!\bmod\mathfrak{w}).$$ 
Hopefully this is helpful to you. If not, all I can suggest is to find a library close to you with access to MathSciNet, so that you can look at the full review yourself. 
