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Abstract
Let f be a non-CM newform of weight k >= 2 without any nontrivial inner twists. In this article we study the set of primes p such that the eigenvalue a_{p}(f) of the Hecke operator T_{p} acting on f generates the field of coefficients of f. We show that this set has density 1, and prove a natural analogue for newforms having inner twists. We also present some new data on reducibility of Hecke polynomials, which suggest questions for further investigation. |