Arithmetic structure of $L_2$-norms of ${\mathrm{SL}}_2(\mathbb{Z})$ matrices

For a matrix $γ\in\mathrm{SL}_2({\mathbb Z})$, we define \[ {\mathcal R}(γ)=a_1^2+a_2^2+a_3^2+a_4^2, \qquad \text{where} \ γ=\begin{pmatrix}a_1&a_2\\ a_3&a_4\end{pmatrix}, \] and let $S_{\mathrm{sq}}(X)$ count the number of matrices $γ$ with $\|γ\|_\infty = \max\{|a_1|, |a_2|,|a_3|,|a_4|\} \leq X$ a