The discriminant formula for the invariant εΓ(X)

Let $\Gamma$ be a Fuchsian group, let $\Sigma$ be a double coset of $\Gamma$, and let $X$ be a $\Gamma$-conjugacy class contained in $\Sigma$. For $M\in X$, define the discriminant of the associated quadratic form by

$$\Delta(M)=\operatorname{tr}^2 M-4\det M,$$

and let $\Gamma_M$ be the centralizer of $M$ in $\Gamma$. Let $\chi(\Gamma)$ be the Euler characteristic defined in the source. The discriminant formula. For any representative $M\in X$,

$$\varepsilon_\Gamma(X)=\begin{cases} -\dfrac{\operatorname{sgn}\Delta(M)}{|\Gamma_M|}\\ -\chi(\Gamma) \end{cases}$$

where the cases are as specified by the source. This conjectural formula would give an explicit description of the invariants entering the trace formula for Hecke operators on Fuchsian groups; the supplied text does not state whether it has been proved or remains open.