G-subsets and G-orbits of Q(sqrt n) under action of the modular group

It is well known that $G=langle x,y:x^2=y^3=1rangle$ represents the modular group $PSL(2,Z)$, where $x:zrightarrowfrac{-1}{z}, y:zrightarrowfrac{z-1}{z}$ are linear fractional transformations. Let $n=k^2m$, where $k$ is any non zero integer and $m$ is square free positive integer. Then the set $$Q^*(sqrt{n}):={frac{a+sqrt{n}}{c}:a,c,b=frac{a^2-n}{c}in Z~textmd{and}~(a,b,c)=1}$$ is a $G$-subset of the real quadratic field $Q(sqrt{m})$ cite{R9}. We denote $alpha=frac{a+sqrt{n}}{c}$ in $ Q^*(sqrt{n})$ by $alpha(a,b,c)$. For a fixed integer $s>1$, we say that two elements $alpha(a,b,c)$, $alpha(a,b,c)$ of $Q^*(sqrt{n})$ are $s$-equivalent if and only if $aequiv a(mod~s)$, $bequiv b(mod~s)$ and $cequiv c(mod~s)$. The class $[a,b,c](mod~s)$ contains all $s$-equivalent elements of $Q^*(sqrt{n})$ and $E^n_s$ denotes the set consisting of all such classes of the form $[a,b,c](mod~s)$. In this paper we investigate proper $G$-subsets and $G$-orbits of the set $Q^*(sqrt{n})$ under the action of Modular Group $G$