Number of proportions of Hecke eigenvalues in two newforms

Abstract

Let f1 and f2 be nonzero Hecke eigenforms on SL2(Z). For X>0, let B(f1,f2,X) be the number of ratios of λ1(p) to λ2(p) for primes p≤X, where λi(p) denotes the Hecke eigenvalue of fi at p. In this paper, we prove that if f1 is not a constant multiple of f2, thenB(f1,f2,X)≫f1,f2(log⁡X)115.