The Second Moment of Sums of Hecke Eigenvalues I

Let $f$ be a holomorphic Hecke cusp form of weight $k$ for ${\mathrm{S}\mathrm{L}}_2(\mathbb{Z})$, and let $({\lambda }_f(n){)}_{n\ge 1}$ denote its sequence of Hecke eigenvalues. We compute the first and second moments of the sums $S(x,f)={\sum }_{x\le n\le 2x}{\lambda }_f(n)$, on average over forms $f$ of large weight $k$, in the regime where the length of the sums $x$ is smaller than $k^2$. We observe transitions in the size of the sums when $x\approx k$ and $x\approx k^2$. In subsequent work (part II), it will be shown that once $x$ is larger than $k^2$ (where the latter transition occurs), the average size of the sums $S(x,f)$ becomes dramatically smaller.