show that $f(x)=a^2 + b^2 + c^2 + d^2$ is smooth on $\text{SL}_2(\mathbb{Z})\backslash \text{SL}_2(\mathbb{R})$

Question

The space of lattices in the Euclidean plane $X=\text{SL}_2(\mathbb{Z})\backslash \text{SL}_2(\mathbb{R})$ can it have smooth functions? I'm trying to find examples of $C^\infty$ functions. Consider equivalence classes of matrices (lattices) $$ f: \left\{ x=\left( \begin{array}{cc} a & b \\ c & d \end{array} \right) : ad-bc=1\right\} \mapsto a^2 + b^2 + c^2 + d^2\in \mathbb{R}$$ Can we show this function is smooth? $f(x)\in C^\infty(X)$