QUADRATIC MINIMA AND MODULAR-FORMS

We give upper bounds on the size of the gap between the constant term and the next nonzero Fourier coefficient of an entire modular form of given weight for inverted right perpendicular (0)(2). Numerical eviden ce indicates that a sharper bound holds for the weights h equivalent t o 2 (mod 4). We derive upper bounds for the minimum positive integer r epresented by level-two even positive-definite quadratic forms. Our da ta suggest that, for certain meromorphic modular forms and p = 2, 3, t he p-order of the constant term is related to the base-p expansion of the order of the pole at infinity.