Arch. Math. 109 (2017), 223–229
2017 Springer International Publishing AG
published online July 8, 2017
Archiv der Mathematik
Comparing Hecke eigenvalues of newforms
Abstract. Given two distinct newforms with real Fourier coeﬃcients, we
show that the set of primes where the Hecke eigenvalues of one of them
dominate the Hecke eigenvalues of the other has density ≥1/16. Fur-
thermore, if the two newforms do not have complex multiplication, and
neither is a quadratic twist of the other, we also prove a similar result for
the squares of their Hecke eigenvalues.
Mathematics Subject Classiﬁcation. 11F30, 11F11.
Keywords. Hecke eigenvalues of newforms, Symmetric power L-functions,
Cuspidal automorphic representations.
1. Introduction. Let k ≥ 2 be an even integer and N a positive integer.
Denote by S
the space of all cuspidal newforms of weight k, for the
congruence subgroup Γ
(N) ⊆ SL
(Z), with trivial Nebentypus. Any f ∈
has a Fourier expansion at inﬁnity in the upper-half plane (z) > 0
The fact that f has trivial Nebentypus implies that the number ﬁeld K
ated by all the Fourier coeﬃcients a
(n) is totally real. We ﬁx a real embedding
and note that our analysis below is independent of this choice. Since f
is a newform, it is a simultaneous eigenform for all the Hecke operators, with
the corresponding Hecke eigenvalues a
(n). For convenience, we shall consider
the normalized eigenvalues
The extent to which the signs of λ
(p)atprimesp determine f uniquely has
been ﬁrst studied by Kowalski et al.  (and also by Matom¨aki , who reﬁned
some of their results). We shall concern ourselves with the following related