Modular forms which behave like theta series

In this paper, we determine all modular forms of weights $36\leq k\leq 56$, $4\mid k$, for the full modular group $SL_2(\mathbb Z)$ 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 ${\mathbb R}^3$ (resp. in ${\mathbb R}^4$) for the cases $k = 36, 40 \hbox {~and~} 44$ (resp. for the cases $k = 48, 52 \hbox {~and~} 56$). 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.