An explicit formula for the number of classes of primitive hyperbolic elements in the group $\Gamma_0(N)$

An explicit formula expressing the number of classes of primitive hyperbolic elements in the congruence subgroup $\Gamma_0(N)$ (the number of closed geodesics) in terms of the number of equivalence classes of indefinite binary quadratic forms is obtained. The well-known formulae for the numbers of classes of elliptic and parabolic elements in $\Gamma_0(N)$ are special cases of this formula.
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