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 χ a Dirichlet character of p-power order and sufficiently large p-power conductor. We show that the compositum Qf(χ) of the Hecke fields associated to f and χ is generated by the square of the absolute value of the corresponding central L-value Lᵃˡᵍ(1/2, f ⊗ χ) over Q(μₚ). 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.