The Hecke group $H({\lambda }_4)$ acting on imaginary quadratic number fields

Let $H({\lambda }_4)$ be the Hecke group $⟨x,y:x^2=y^4=1⟩$ and, for a square-free positive integer $n$, consider the subset $$\mathbb{Q}$^*(\sqrt{-n})=\left{(a+\sqrt{-n})/c , | , a,b=(a^2+n)/c \in $\mathbb{Z}$,, c\in 2$\mathbb{Z}$ \right}$ of the quadratic imaginary number field $\mathbb{Q}(\sqrt{-n})$. Following a line of research in the relevant literature, we study properties of the action of $H({\lambda }_4)$ on ${\mathbb{Q}}^{\ast }(\sqrt{-n})$. In particular, we calculate the number of orbits arising from this action for every such $n$. Some illustrative examples are also given.