Multiplicative characteristics of function for the number of classes of primitive hyperbolic elements in the group $\Gamma_0(N)$ by level $N$

An arithmetical forms of Selberg's trace formula and Selberg's zeta-function for the congruence subgroup $\Gamma_0(N)$, explicit expression for the number of classes of primitive hyperbolic elements in the congruence subgroup level $N$ in terms of the number of classes of primitive elements in the congruence subgroup level $N_1=N/P^i$, $(N,N_1)=1$ and sharp upper bound of the number classes by level $N$ are obtained.