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TITLE: [GS_M_NT] Generation of Hecke fields by the square of absolute values of modular $L$-values with cyclotomic twists

ABSTRACT: Let $f$ be a non-CM elliptic newform, which does not have a quadratic inner twist. Let $p$ be an odd prime and $\chi$ a $p$-power conductor Dirichlet character. 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 $L^{alg}(1/2, f \otimes \chi)$ over $\mathbb{Q}(\mu_p)$, as $\chi$ varies over Dirichlet characters of $p$-power conductor and order. The proof is based on the recent resolution of unipotent mixing conjecture due to Blomer and Michel.