Bounds for coefficients of cusp forms and extremal lattices

Abstract

A cusp form $f(z)$ of weight $k$ for $\SL_{2}(\Z)$ is determined uniquely by its first $\ell :=\dim S_k$ Fourier coefficients. We derive an explicit bound on the $n$th coefficient of $f$ in terms of its first $\ell$ coefficients. We use this result to study the non-negativity of the coefficients of the unique modular form of weight $k$ with Fourier expansion \[F_{k,0}(z) = 1 + O(q^{\ell + 1}).\] In particular, we show that $k=81632$ is the largest weight for which all the coefficients of $F_{0,k}(z)$ are non-negative. This result has applications to the theory of extremal lattices.

To appear in Bulletin of the London Mathematical Society

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