First I am considering the function $\tilde{f}$ : $ SL(2, \mathbb{R}) \rightarrow \mathbb{C}$ such that $\tilde{f} = (ci + d)^{-m}f(gi)$.
If $r_k$ with $k \in S0(2,\mathbb{R})$ denotes that right translation map $r_k \tilde{f}(g) = \tilde{f}(gk)$ $\forall g,k \in SL(2, \mathbb{R})$ then the set $V =$ { $r_k \tilde{f} | k \in S0(2,\mathbb{R})$ } is a one dimensional vector space generated by $ \tilde{f}$.
I am not entirely sure how to prove that this set $V$ is a vector space generated by $ \tilde{f}$.