Determination of a pair of newforms from the product of their twisted central values

We show that a pair of newforms $(f,g)$ can be uniquely determined by the product of the central $L$-values of their twists. To achieve our goal, we prove an asymptotic formula for the average of the product of the central values of two twisted $L$-functions- $L(1\mathrm{/}2,f\times \chi )L(1\mathrm{/}2,g\times \chi \psi )$, where $(f,g)$ is a pair of newforms. The average is taken over the primitive Dirichlet characters $\chi$ and $\psi$ of distinct prime moduli.