Generation of Hecke fields by squares of cyclotomic twists of modular $L$-values

Abstract: Let $f$ be a non-CM elliptic newform without a quadratic inner twist, $p$ an odd prime and $\chi$ a Dirichlet character of $p$-power order and sufficiently large $p$-power conductor. We show that the compositum $\mathbb{Q}_{f}(\chi)$ of the Hecke fields associated to $f$ and $\chi$ is generated by the square of the absolute value of the corresponding central $L$-value $L^{\rm alg}(1/2, f \otimes \chi)$ over $\mathbb{Q}(\mu_p)$. The proof is based among other things on techniques used for the recent resolution of unipotent mixing conjecture by the first and third named authors.