  We use recent results of Rolen, Zwegers, and the first author to study
characters of irreducible (highest weight) modules for the vertex operator
algebra $L_{\frak{sl}_\ell}(-\Lambda_0)$. We establish asymptotic behaviors of
characters for the (ordinary) irreducible
$L_{\frak{sl}_\ell}(-\Lambda_0)$-modules. As a consequence we prove that their
quantum dimensions are one, as predicted by representation theory. We also
establish a full asymptotic expansion of irreducible characters for
$\frak{sl}_3$. Finally, we determine a decomposition formula for the full
characters in terms of unary theta and false theta functions which allows us to
study their modular properties.
