  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.
