Bounds for coefficients of cusp forms and extremal lattices

A cusp form $f(z)$ of weight $k$ for ${\text{SL}}_2(\mathbb{Z})$ is determined uniquely by its first $\mathrm{\ell }:=\dim S_k$ Fourier coefficients. We derive an explicit bound on the $n$th coefficient of $f$ in terms of its first $\mathrm{\ell }$ coefficients. We use this result to study the non-negativity of the coefficients of the unique modular form of weight $k$ with Fourier expansion $$F_{k,0}(z)=1+O(q^{\mathrm{\ell }+1})\mathrm{.}$$ 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.