Let f be a non-CM newform of weight k ≥ 2. 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.Mathematics Subject Classification (2000): 11F30 (primary); 11F11, 11F25, 11F80, 11R45 (secondary).

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The goal of this note is to prove, under some assumptions, a formula relating the Selmer groups of isogenous Galois representations. Local and global Euler-Poincaré characteristic formulas are key tools in the proof. With additional hypotheses, we use the isogeny formula to study how the formation of Selmer groups interacts with normalization of the coefficient ring and discuss how a main conjecture for a big Galois representation over a non-normal ring follows from a corresponding conjecture over the normalization.

It is a classical result that the number of primes for which τ ( ) vanishes has Dirichlet density 0, where τ ( ) is the Ramanujan τ function. We study an analogous question that arises in studying the Λ-adic representation whose image is full. In particular, we show that the set of primes for which the trace of the Frobenius at has positive μ-invariant has Dirichlet density 0. We also discuss the analogous Dirichlet densities related to λ-invariants.