Let f,g be newforms. For odd prime p, we let $\chi$ be a primitive Dirichlet character of p-power conductor and order. Due to the work of Shimura, it is known that there exist algebraic modular L-values $L_f(\chi)$ and $L_g(\chi)$. We prove that the Hecke field $Q_fQ_g(\chi)$ is generated by the products $L_f(\chi)L_g(\chi)$, for χ with large conductor. This result was derived by estimating a Galois average analogue of the second moment of central modular L-values, due to Blomer-Mili\'cevi\'c. This is a joint work with Ashay Burungale, Philippe Michel and Valentin Blomer.