Restricted representations of the twisted $N=2$ superconformal algebra

In this paper, we construct a large class of new simple modules over the twisted $N=2$ superconformal algebra. These new simple modules are restricted modules based on the simple modules over certain finite-dimensional solvable Lie superalgebras, including various versions of Whittaker modules. We elaborate that they are also the twisted modules for the universal $N=2$ superconformal vertex algebra. On the other hand, we give an explicit characterization of the simple restricted modules over the twisted $N=2$ superconformal algebra $\mathcal{T}$ under the condition that $T_t$ in $\mathcal{T}$ acts injectively for some $t\in \frac{1}{2}+{\mathbb{Z}}_{+}$.