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

Abstract: For square-free positive integers $n$, we study the action of the modular group $\mbox{PSL}(2,\mathbb{Z})$ 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 $1\leq n \leq 100$ are also given.