I am trying to work out $2$-dimensional representation of $\operatorname{SL}_2(\mathbb{Z})$. I know that $\operatorname{SL}_2(\mathbb{Z})$ is generated by $S = \begin{pmatrix} 0 & -1\\ 1 & 0 \end{pmatrix}, R = \begin{pmatrix} 0 & -1\\ 1 & 1\end{pmatrix}$ and $T= \begin{pmatrix} 1 & 1\\ 0 & 1 \end{pmatrix}$.
My lecture notes say that for $\rho\colon\operatorname{SL}_2(\mathbb{Z}) \rightarrow \operatorname{GL}_2(\mathbb{C})$, I should diagonalize $\rho(S)$ and see what choices I have for $\rho(R)$. But I am not sure how to work these calculations out. Can someone help? Thank you in advance!
$\mathrm{SL}_2(\mathbb Z)$ is generated by $S$ and $R$ and we have $S^4 = \mathrm{id}$ and $S^2 = R^3 = -\mathrm{id}$. By diagonalizing $\rho$, we can assume, that $\rho(S) = \begin{pmatrix} \alpha & 0 \\ 0 & \beta\end{pmatrix}$ for $\alpha, \beta \in \{\pm 1, \pm i\}$. We have $R^6 =\mathrm{id}$, so $\rho(R)$ is diagonalizable with eigenvalues being $\gamma,\delta \in \{1,\zeta, \dots, \zeta^5\}$, where $\zeta = e^{\frac{2\pi i}{6}}$. Hence $\rho(R) = A\begin{pmatrix} \gamma & 0 \\ 0 & \delta\end{pmatrix}A^{-1}$ for some $A \in \mathrm{GL}_2(\mathbb C)$.