  We give upper bounds on the size of the gap between the constant term and the
next non-zero Fourier coefficient of an entire modular form of given weight for
\Gamma_0(2). Numerical evidence indicates that a sharper bound holds for the
weights h \equiv 2 . We derive upper bounds for the minimum positive integer
represented by level two even positive-definite quadratic forms. Our data
suggest that, for certain meromorphic modular forms and p=2,3, the p-order of
the constant term is related to the base-p expansion of the order of the pole
at infinity, and they suggest a connection between divisibility properties of
the Ramanujan tau function and those of the Fourier coefficients of 1/j.
