The second moment of sums of Hecke eigenvalues II

Abstract Let be a holomorphic Hecke cusp form of weight for , and let denote its sequence of normalised Hecke eigenvalues. We compute the first and second moments of the sums , on average over forms of large weight . In the range , the size of the second moment lies between and . This is in sharp contrast to the regime , where the second moment was shown in preceding work (part I) to be of size .