I have encountered the following exponential sum and I would like to obtain a non-trivial upper bound for it. I am not quite sure where to start, and I would greatly appreciate any suggestions on how I can go about to obtain an upper bound or where to look.

Let $q \in \mathbb{N}$, and $\chi_i$ be a Dirichlet character modulo $q$ (not necessarily primitive), and $F(x_1, \ldots, x_n)$ is a homogeneous polynomial of degree $d$ with integer coefficients which defines a smooth hypersurface over $\mathbb{P}^{n-1}$.

The exponential sum in question is $$ \sum_{\mathbf{x} \in (\mathbb{Z}/q \mathbb{Z})^n } \chi_1(x_1)\cdots \chi_n(x_n) \ e \left(F(\mathbf{x}) \frac{a}{q} \right), $$ where $(a,q)=1$.

I would greatly appreciate any suggestions or comments. Thank you very much for your time.