Let $G$ be the principal congruence subgroup of level $N \geq 3$ and $g$ be the group generated by the involution $z \mapsto -1/z$ of the upper half plane. We shall determine the cardinality of the (first) cohomology set $H(g,G)$ in terms of the binary form $x^2 + y^2 \mod N$.
Takashi Ono. "On certain cohomology sets attached to Riemann surfaces." Proc. Japan Acad. Ser. A Math. Sci. 76 (7) 116 - 117, Sept. 2000. https://doi.org/10.3792/pjaa.76.116