Geodesic
An explicit formula for the number of classes of primitive hyperbolic elements in the group $\Gamma_0(N)$
Sbornik. Mathematics, Tome 199 (2008) no. 7, pp. 1009-1031 Cet article a éte moissonné depuis la source Math-Net.Ru

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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. Bibliography: 11 titles. 
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V. V. Golovchanskii; M. N. Smotrov. An explicit formula for the number of classes of primitive hyperbolic elements in the group $\Gamma_0(N)$. Sbornik. Mathematics, Tome 199 (2008) no. 7, pp. 1009-1031. http://geodesic.mathdoc.fr/item/SM_2008_199_7_a3/

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