  Duke, Imamo\={g}lu, and T\'oth have recently constructed a new geometric
invariant, a hyperbolic orbifold, associated to each narrow ideal class of a
real quadratic field. Furthermore, they have shown that the projection of these
hyperbolic orbifolds onto the modular surface $\Gamma \backslash \mathbb{H}$
equidistributes on average over a genus of the narrow class group as the
fundamental discriminant $D$ of the real quadratic field tends to infinity.
  We extend this construction of hyperbolic orbifolds to allow for a level
structure, akin to Heegner points and closed geodesics of level $q$.
Additionally, we refine this equidistribution result in several directions.
First, we investigate sparse equidistribution in the level aspect, where we
prove the equidistribution of level $q$ hyperbolic orbifolds when restricted to
a translate of $\Gamma \backslash \mathbb{H}$ in $\Gamma_0(q) \backslash
\mathbb{H}$, which presents some new interesting features. Second, we explore
sparse equidistribution in the subgroup aspect, namely equidistribution on
average over small subgroups of the narrow class group. Third, we prove small
scale equidistribution and give upper bounds for the discrepancy.
  Behind these refinements is a new interpretation of the Weyl sums arising in
these equidistribution problems in terms of ad\`elic period integrals, which in
turn are related to Rankin-Selberg $L$-functions via Waldspurger's formula. The
key remaining inputs are hybrid subconvex bounds for these $L$-functions and a
certain homological version of the sup-norm problem.
