Geometry and distribution of roots of ${\mu }^2\equiv D<mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d</mi></mrow>m$ with $D\equiv 1<mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d</mi></mrow>4$

We extend the geometric interpretation of the roots of the quadratic congruence ${\mu }^2\equiv D(\mathrm{m}\mathrm{o}\mathrm{d}m)$ as the tops of certain geodesics in the hyperbolic plane to the case when $D$ is square-free and $D\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}4)$. This allows us to establish limit laws for the fine-scale statistics of the roots using results by Marklof and the second author.