Orbits of ${\text{QQ}}^{\ast }(\sqrt{n})$ under the action of Modular Group $PSL(2,\mathbb{Z})$

Coset diagrams have been used to study quotients, orbits, subgroups and structure of the finitely generated groups. In this paper we use coset diagrams and modular arithmetic to determine the $G$-orbits of ${\text{QQ}}^{\ast }(\sqrt{p^k})$, ${\text{QQ}}^{\ast }(\sqrt{2p^k})$, ${\text{QQ}}^{\ast }(\sqrt{2^2{p}^k})$, and in general ${\text{QQ}}^{\ast }(\sqrt{2^l{p}^k})$, for each $l\ge 3$ and $k=2h+1\ge 3$, for each odd prime $p$.