Topological modular forms with level structure: decompositions and duality

Topological modular forms with level structure were introduced in full generality by Hill and Lawson. We show that these decompose additively in many cases into a few simple pieces and give an application to equivariant $TMF$. Furthermore, we show which $Tm{f}_1(n)$ are self-Anderson dual up to a shift, both with and without their natural $C_2$-action.