Modular invariant representations of the $\mathcal{N}=2$ superconformal algebra

We compute the modular transformation formula of the characters for a certain family of (finitely or uncountably many) simple modules over the simple $\mathcal{N}=2$ vertex operator superalgebra of central charge $c_{p,p'}=3\left(1-\frac{2p'}{p}\right),$ where $(p,p')$ is a pair of coprime positive integers such that $p\geq2$. When $p'=1$, the formula coincides with that of the $\mathcal{N}=2$ unitary minimal series found by F. Ravanini and S.-K. Yang. In addition, we study the properties of the corresponding "modular $S$-matrix", which is no longer a matrix if $p'\geq2$.