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

Koopa Koo
William Stein
Gabor Wiese

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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 ap(f) of the Hecke operator Tp 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.