Modified Jacobi forms of index zero (II)

For a negative integer $k$ let $J_k$ be the space of modified Jacobi forms of weight $k$ and index 0 on ${\mathrm{S}\mathrm{L}}_2(\mathbb{Z})$. For each positive integer $m$ we consider certain subspace $J_k^m$ of $J_k$ which satisfies $J_k={\cup }_{m=1}^{\mathrm{\infty }}{J}_k^m$. By observing a relation between coefficients of the Fourier development of a modified Jacobi form we show that $J_k^m$ is finite-dimensional.