8.82 (1982)

Let $\mathfrak{H} = \mathbb{C} \times \mathbb{R} = \{(z, r) \mid z \in \mathbb{C}, \ r > 0\}$ be the three-dimensional Poincaré space which admits the following action of the group $SL_2(\mathbb{C})$:
$$(z,r) \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \left( \frac{(az+b)(\overline{cz+d}) + a\bar{c}r^2}{|cz+d|^2 + |c|^2r^2}, \ \frac{r}{|cz+d|^2 + |c|^2r^2} \right).$$ Let $\mathfrak{o}$ be the ring of integers of the field $K = \mathbb{Q}(\sqrt{D})$, where $D < 0$, $\pi$ a prime such that $\pi\overline{\pi}$ is a prime in $\mathbb{Z}$ and let
$$\Gamma = \left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in SL_2(\mathfrak{o}) \ \middle|\ b \equiv 0 \pmod{\pi} \right\}.$$ Adding to the space $\Gamma \backslash \mathfrak{H}^3$ two vertices we get a three-dimensional compact space $\overline{\Gamma \backslash \mathfrak{H}^3}$.

Calculate
$$r(\pi) = \operatorname{dim}_\mathbb{Q} H_1(\overline{\Gamma \backslash \mathfrak{H}^3}, \mathbb{Q}) = \operatorname{dim}_\mathbb{Q}(\Gamma^{ab} \otimes \mathbb{Q}).$$

For example, if $D = -3$ then $r(\pi)$ is distinct from zero for the first time for $\pi \mid 73$ (then $r(\pi) = 1$), and if $D = -4$ then $r(\pi) = 1$ for $\pi \mid 137$.