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{\chi\ primitive\ cubic\\ genus(\chi)=g}}\chi(h_1)\bar{\chi}(h_2)|L_q(\frac{1}{2}, \chi)|^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