  Among topological modular forms with level structure, $TMF_0(7)$ at the prime
$3$ is the first example that had not been understood yet. We provide a
splitting of $TMF_0(7)$ at the prime 3 as $TMF$-module into two shifted copies
of $TMF$ and two shifted copies of $TMF_1(2)$. This gives evidence to a much
more general splitting conjecture. Along the way, we develop several new
results on the algebraic side. For example, we show the normality of rings of
modular forms of level $n$ and introduce cubical versions of moduli stacks of
elliptic curves with level structure.
