Is there a natural example of a discrete subgroup $\Gamma\leq PSL_2(\mathbf{R})$ such that

(1) $\Gamma$ has finite covolume

(2) $\mathfrak{h}/\Gamma$ is not compact ($\mathfrak{h}$ being the upper half-plane)

(3) $\Gamma$ is not commensurable to a conjugate of $PSL_2(\mathbf{Z})$.

I cannot think of any such example but I don't see any reason why they should not exist.