On certain partial sums involving squares of Hecke eigenvalues

Let $a(n)$ be the nth Fourier coefficient of a cuspidal Hecke eigenform of even integral weight k≥ 2 and trivial character that is a normalized new form for some level N. We show that the partial sumsHₙ=∑ₘ₌₁ⁿ a(m)²/mᵏare not integral for n≥ n₀.