Let $\Gamma = PSL_2(\mathbb{Z}) $ and f be a $\Gamma$ invariant function on the upper half plane. Further let $F$ be the standard fundamental domain of $\Gamma$ and $F^{’ }=gF$ be the image of $F$ under any fixed $g \in \Gamma$. Define a new function $\tilde{f}(h) =f(h)$ if $ h \in F^{'}$ otherwise its zero.
How can we conclude that $f(h)= \sum_{g \in \Gamma}{} \tilde{f}(gh) $ with only a discrete set of exceptions.
My idea: I split it into two cases, one is when $h \in F^{’}$ and other is when $h \notin F^{’}$. In the first case using the fact that $gh \notin F^{’}$ for any g other than identity I will get the required equality. But for the other case I am not able to get that how can I write my f as that sum with some discrete set of exceptions. Any help would be appreciated.
Thanks in advance.