Generation of cyclotomic Hecke fields by $L$-values of cusp forms on $\mathrm{GL}(2)$ with certain $\mathbb{Z}_p$ twist

Let $F$ be a number field, $f$ an algebraic automorphic newform on $\mathrm{GL}(2)$ over $F$, $p$ an odd prime does not divide the class number of $F$ and the level of $f$. We prove that $f$ is determined by its $L$-values twisted by Galois characters $\phi$ of certain $\mathbb{Z}_p$-extension of $F$. Furthermore, if $F$ is totally real or CM, then under some mild assumption on $f$, the compositum of the Hecke field of $f$ and the cyclotomic field $\mathbb{Q}(\phi)$ is generated by the algebraic $L$-values of $f$ twisted by Galois characters $\phi$ of certain $\mathbb{Z}_p$-extension of $F$.