simple cartan-free modules of finite rank over the super-virasoro algebras

We classify nontrivial simple super-Virasoro modules finitely generated over $\text{C}[L_0]$ using simple Laurent Weyl algebra modules finite free over $\text{C}[D]$. The classified modules are finite Cartan-free with equal parity ranks. We also prove finite length without simplicity and construct explicit nonsplit self-extensions.