Explicit formula for sums related to the generalized Bernoulli numbers

Let $\chi$ be a Dirichlet character modulo a prime number $p\geqslant 3$ and let $B_m(\chi)$ $(m=1,2,\ldots)$ be the generalized Bernoulli numbers associated with $\chi$. Explicit formulas for the sums:
$$\sum_{\substack{\chi\mod p\\\chi(-1)=+1, \chi\neq\chi_0}}B_{m}(\chi)B_{n}(\overline{\chi})\text{ and }\sum_{\substack{\chi\mod p\\ \chi(-1)=-1}}B_{m}(\chi)B_{n}(\overline{\chi})$$
are given in this paper.