Modular forms and almost linear dependence of graded dimensions

For every positive integral level $k$ we study arithmetic properties of certain holomorphic modular forms associated to modular invariant spaces spanned by graded dimensions of $L_{\hat{sl_2}}(k \Lambda_0)$-modules. We found a necessary and sufficient condition for their vanishing and showed that these modular forms resemble classical Eisenstein series $E_{2k+2}(\tau)$. Furthermore, we derived similar results for $\mathcal{M}(p,p')$ Virasoro minimal models, thus generalizing some results of Mortenson, Ono and the author.