Maybe someone could help me out. I consider a $SL_2(\mathbb{Z})$-module $\Omega$. We set \begin{align} S:=\left(\begin{array}{rr} 0 & 1\\ -1 & 0\end{array}\right) \ \mathrm{and} \ U:=\left(\begin{array}{rr} 0 & 1\\ -1 & 1\end{array}\right). \end{align} I want to compute the quotient \begin{align} \Omega/\left((1+S) \ \Omega \oplus (1+U+U^2) \ \Omega\right). \end{align} Trivially in this quotient the so called two and three term relations \begin{align} \omega + S\omega=0 \ \mathrm{and} \ \omega+U\omega+U^2\omega=0 \end{align} hold. So I am wondering if there is maybe a connection to the group of $\Omega$-valued $SL_2(\mathbb{Z})$-invariant modular symbols or something in this direction?
Thanks for granted :)
Edit: Maybe I should be more precise: The module I consider is (more or less) the free module generated by $\mathbb{P}^1(\mathbb{Q})$ and $SL_2(\mathbb{Z})$-acts via Möbius transformations.