Random ordering in modulus of consecutive Hecke eigenvalues of primitive forms

Abstract

Let \tau(.) be the Ramanujan \tau-function, and let k be a positive integer such that \tau(n) is not 0 for n=1,...,[k/2]. (This is known to be true for k < 10^{23}, and, conjecturally, for all k.) Further, let s be a permutation of the set {1,...,k}. Then there exist infinitely many positive integers m such that |\tau(m+s(1))|<\tau(m+s(2))|<...<|\tau(m+s(k))|. We also obtain a similar result for Fourier-coefficients of general newforms.