  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 such
that the p-th coefficient of f generates the field L. This density is
determined by the inner twists of f. As a particular case, we obtain that in
the absence of non-trivial inner twists, the density is 1 for L equal to the
whole coefficient field. We also present some new data on reducibility of Hecke
polynomials, which suggest questions for further investigation.
