  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.
