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- $L(1/2, f \times \chi)L(1/2, g \times \chi \psi)$, where $(f,g)$ is a pair of newforms. The average is taken over the primitive Dirichlet characters $\chi$ and $\psi$ of distinct prime moduli.
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