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

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 $L^{\mathrm{a}\mathrm{l}\mathrm{g}}(1\mathrm{/}2,f\otimes \chi )$ 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.