I recently came upon the following kind of sum and I'm wondering if anyone has seen it before, or could point out something interesting about them.

Let $F$ be a finite field with $q > 2$ elements and let $\chi_N$ be a multiplicative character of $F$ with order exactly $N \mid (q-1)$, $N > 1$. Let $\psi$ be a non-trivial multiplicative character of $F$ with order coprime to $N$. The sum I'm interested in is $$ S_N(\psi) = \dfrac{1}{q N} \sum_{j=1}^{N-1} G(\chi_N^j) \overline{J(\chi_N^j, \psi)}, $$ where $G(\chi_N^j)$ and $J(\chi_N^j, \psi)$ are Gaussian and Jacobian sums, respectively. See here for their definition and properties:


I wonder if anyone has seen this before? In particular what is $|S_N(\psi)|$?


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