I am wondering if there are any references that would help me with the following problem:

Let $p>2$ be a prime number, $n \in \mathbb{Z}^+$ odd such that $(n,p)=1$, $\chi$ an imaginary quadratic character of conductor $n$, and $\omega$ the Teichmuller character.

I'm trying to calculate the sum \begin{equation} \sum_{a=1}^{(np-1)/2} \chi\omega^{-1}(a)(-1)^a. \end{equation}

This sum almost reminds me of a quadratic Gauss sum (I know this is a big stretch) which I've seen can be computed using Fourier analysis. From my basic understanding (see this article), the computation relies on the fact that \begin{equation} \sum_{a=0}^{p-1}\left(\frac{a}{p}\right)e^{2\pi i a/p} = \sum_{a=0}^{p-1} e^{2\pi i a^2/p} \end{equation} (here $(a/p)$ is the Legendre symbol) and defining $f(x) = \sum_{a=0}^{p-1} e^{2\pi i (a+x)^2/p}$, which is periodic on $\mathbb{R}$, we compute $\hat{f}(x)$ and find that \begin{equation} \sum_{a=0}^{p-1}\left(\frac{a}{p}\right)e^{2\pi i a/p}= \sum_{a = -\infty}^{\infty}\hat{f}(a) = \text{ a computable integral}. \end{equation}

In my naive attempt to mimic this process, I think I fail (or just can't see) how to write $\sum_{a=1}^{(np-1)/2} \chi\omega^{-1}(a)(-1)^a$ as a sum involving only exponentials and then coming up with a periodic function $f(x)$ that is my character sum when evaluated at 0. Because it is quite a stretch to compare my sum to a Gauss sum, I'm now thinking it is a bad idea to try to mimic this computation.

Questions:

• Is there a more general method out there for computing character sums like the one I have using Fourier analysis?

• Are there any other methods/theory that might help me out in calculating this sum?

I would also be happy if I could somehow factor this sum in $\mathbb{Q}(e^{2\pi i/(p-1)})$. I'm reminded of Stickelberger's theorem, where part of the proof involves finding the prime factorization of a certain Gauss sum (here I go again with the comparison). I know that the sum I'm looking at will not behave as nicely as the Gauss sum in the proof of Stickelberger's, but I am wondering,

• Is there is a general theory for finding the $\mathcal{P}$-adic valuation of a character sum, where $\mathcal{P}$ lies above $p$ in whatever field the character sum lives in.

Thanks in advance for any references or help you can give me (even if it's just saying that this is to difficult to solve)!