Geometry and distribution of roots of $\mu ^2\equiv D\ {\rm mod\ } m$ with $D \equiv 1\ {\rm mod\ } 4$

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