### The distribution of Fourier coefficients of cusp forms over sparse sequences

Huixue Lao, Ayyadurai Sankaranarayanan (2014)

Acta Arithmetica

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Let ${\lambda}_{f}\left(n\right)$ be the nth normalized Fourier coefficient of a holomorphic Hecke eigenform $f\left(z\right)\in {S}_{k}\left(\Gamma \right)$. We establish that ${\sum}_{n\le x}{\lambda}_{f}^{2}\left({n}^{j}\right)={c}_{j}x+O\left({x}^{1-2/({(j+1)}^{2}+1)}\right)$ for j = 2,3,4, which improves the previous results. For j = 2, we even establish a better result.