On certain partial sums involving squares of Hecke eigenvalues

Abstract

Let $a(n)$ be the $n$th Fourier coefficient of a cuspidal Hecke eigenform of even integral weight $k\ge 2$ and trivial character that is a normalized new form for some level $N$. We show that the partial sums$$H_n=\sum_{m=1}^n a(m)^2/m^k$$are not integral for $n\ge n_0$.