Bounds for Coefficients of Cusp Forms and Extremal Lattices

A cusp form f(z) of weight k for SL2(Z) is determined uniquely by its first ` := dimSk Fourier coefficients. We derive an explicit bound on the nth coefficient of f in terms of its first ` coefficients. We use this result to study the nonnegativity of the coefficients of the unique modular form of weight k with Fourier expansion Fk,0(z) = 1 + O(q ). In particular, we show that k = 81632 is the largest weight for which all the coefficients of Fk,0(z) are non-negative. This result has applications to the theory of extremal lattices.