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Let f be a non-CM newform of weight k > 1.  Let L be a subfield of the coefficient field of f.  We completely settle the question of the density of the set of primes p suchthat the p-th coefficient of f generates the field L. This density isdetermined by the inner twists of f. As a particular case, we obtain that inthe absence of non-trivial inner twists, the density is 1 for L equal to thewhole coefficient field. We also present some new data on reducibility of Heckepolynomials, which suggest questions for further investigation.

On the generation of the coefficient field of a newform by a single Hecke eigenvalue