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

Abstract

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).

Newforms and spectral multiplicity for Γ0 (9)

“…For a precise definition and results related to inner twists see e.g. [51,52,40,35]. For the sake of clarity we give a simple example of the situation for holomorphic modular forms.…”

On the Hecke fields of Galois representations

“…If ρ comes from a newform f = ∞ n=1 a n q n of weight 2 that does not have complex multiplication nor any non-trivial inner twists, then Im(ρ) has connected Zariskiclosure (at least for all but finitely many ) by Theorem 3.1 of [12] (see also [11]). Thus our Corollary 2.3 implies Corollary 1 of [8]. However, we have not studied any proper subfields of E such as 'Q(a 2 n /χ(n) : (n, N ) = 1)' in [8], and so we cannot recover the main results of [8] in full generality.…”

“…Thus our Corollary 2.3 implies Corollary 1 of [8]. However, we have not studied any proper subfields of E such as 'Q(a 2 n /χ(n) : (n, N ) = 1)' in [8], and so we cannot recover the main results of [8] in full generality.…”

“…If f = a n q n is the Fourier expansion of f , where q = e 2π √ −1z , then the field E f generated over the rational number field Q by a n for all n 1 with (n, N ) = 1 is of finite degree over Q and is called the Hecke field of f . In many cases, it is known that E f is in fact generated by a p for a single prime number p. For instance, in Corollary 1.1 of [8], it is proved that, under a mild condition, E f is generated by a p for a single p in a set of primes of density 1. (Such studies are motivated, at least partly, by Maeda's conjecture [10].)…”

Modular forms applied to the computational inverse Galois problem

“…Remark. The set of primes p with F f = Q(a 2 p /ε f (p)) has density 1 (see [16,Theorem 1]), so one can expect that the condition on λ ′ is almost always satisfied. In any case we do have an embedding…”