Modular forms which behave like theta series

In this paper, we determine all modular forms of weights 36 ≤ k ≤ 56, 4 | k, for the full modular group SL2(Z) which behave like theta series, i.e., which have in their Fourier expansions, the constant term 1 and all other Fourier coefficients are non–negative rational integers. In fact, we give convex regions in R3 (resp. in R4) for the cases k = 36, 40 and 44 (resp. for the cases k = 48, 52 and 56). Corresponding to each lattice point in these regions, we get a modular form with the above property. As an application, we determine the possible exceptions of quadratic forms in the respective dimensions.