Twisted second moment of primitive cubic L-functions

We investigate the mean value of the twisted second moment of primitive cubic $L$-functions over ${\mathbb{F}}_q(T)$ in the non-Kummer setting. Specifically, we study the sum \begin{equation*} \sum_{\substack{χ primitive\ cubic\ genus(χ)=g}}χ(h_1)\barχ(h_2)|L_q(\frac{1}{2}, χ)|^2, \end{equation*} where $L_q(s,\chi )$ denotes the $L$-function associated with primitive cubic character $\chi$. Employing a double Dirichlet series approach, we establish an error term of size