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

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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. 
@article{SM_2008_199_7_a3,
     author = {V. V. Golovchanskii and M. N. Smotrov},
     title = {An explicit formula for the number of classes of primitive hyperbolic elements in the group $\Gamma_0(N)$},
     journal = {Sbornik. Mathematics},
     pages = {1009--1031},
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     number = {7},
     year = {2008},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2008_199_7_a3/}
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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/