One can count hyperbolic conjugacy classes in $\mathrm{SL}_2(\mathbb{Z})$ according to their traces. The result is the prime geodesic theorem, which bears a close similarity with the prime number theorem. As primes are equidistributed in reduced residue classes, the natural question arises if the same is true of the traces mentioned above. It turns out that the answer is no, and the corresponding non-uniform distribution can be determined explicitly. This confirms a conjecture of Golovchanski--Smotrov (1999). Based on joint work with Dimitrios Chatzakos and Ikuya Kaneko.