On the number of orbits arising from the action of $PSL(2,\mathbb{Z})$ on imaginary quadratic number fields

For square-free positive integers $n$, we study the action of the modular group $PSL(2,\mathbb{Z})$ on the subsets ${,\frac{a+\sqrt{-n}}{c}\in $\mathbb{Q}$(\sqrt{-n}), | , a,b=\frac{a^2+n}{c},c \in $\mathbb{Z}$ ,}$ of the imaginary quadratic number fields $\mathbb{Q}(\sqrt{-n})$. In particular, we compute the number of orbits under this action for all such $n$ as provide an interesting congruence property of this number. An illustrative example and a C${}^{++}$ code to calculate such a number for all $1\le n\le 100$ are also given.