On the number of dominating Fourier coefficients of two newforms

Abstract

Let f = n≥1 λ f (n)n (k 1 −1)/2 q n and g = n≥1 λg(n)n (k 2 −1)/2 q n be two newforms with real Fourier coeffcients. If f and g do not have complex multiplication and are not related by a character twist, we prove that #{n ≤ x | λ f (n) > λg(n)} x.

Some Density Results on Sets of Primes for Hecke Eigenvalues

“…Inspired by [13], Chiriac [15] started to compare Hecke eigenvalues over prime numbers and simultaneously showed that the sets of primes for λ f (p) < λ g (p) and λ 2 f (p) < λ 2 g (p) both have analytic density at least 1/16. Notice that the pair-Sato-Tate conjecture yields a stronger result for the former set in [15] with natural density 1/2 in replace of at least 1/16 (see Proposition 2.1 (iii) in [16]). Of course, this result is also valid for the analytic density since the existence of the natural density implies that of the analytic density, and they are equal.…”

“…sym 2l 1 f×sym 2l 2 f (p). (75)Combining the above result with(15) and(16) leads to p O(1), as s ⟶ 1 + . (76)From (15),(17), and (72), it is easy to check that ass ⟶ 1 + , p λ 3 f p j λ g p j p s � (j + 1) p λ sym j f×sym j g (p) p s + p jλ sym 3j− 2i f×sym j g (p) p s + p [j/2] i�1 (j − 2i + 1) λ sym j− 2i f×sym j g (p) p s � O(1).…”