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Let $F$ be a finite field with $q$ elements and characteristic $p$. Let $E$ be a proper extension over $F$ of degree $n$. Let $\psi$ be the canonical additive character of $E$ defined by $\psi(x) = e^{2\pi\sqrt{-1} T(x)/p}$, where $T$ is the (absolute) trace map from $E$ to $\mathbb{F}_p$. For a divisor $N$ of $q^n-1$, a multiplicative character $\chi_N$ of $E$ of order exactly $N$ is given by $\chi_N(\alpha^k) = e^{2\pi\sqrt{-1}jk/N}$, for some $(j, N) = 1$, where $\alpha$ is a primitive element of $E$, i.e., $E^* = \left< \alpha\right>$. The Gauss sum $G(\chi_N)$ associated to $\chi_N$ is given by $$ G(\chi_N) = \sum_{x \in E^*} \chi_N(x) \psi(x). $$

Let $d \mid (q^n-1)/(q-1)$ and $b \mid (q-1)$ with $(b,d) = 1$. Let $\chi_b$ and $\chi_d$ be multiplicative characters of $E$ with orders exactly $b,d$, respectively. Are there any interesting multiplicative properties of $G(\chi_N)$ as a function of $N$ that could allow us to write $G(\chi_b \chi_d)$ in terms of $G(\chi_b)$ and $G(\chi_d)$? Thanks!

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