I am looking for a reference (if one exists) for an application of Selberg's trace formula after the Hecke operators have been applied. Perhaps to give some context and notation to this, so my question is clear. Perhaps, we should just let $\Gamma=\text{SL}(2,\mathbb{Z})$ (or in general a cocompact lattice acting on $\mathbb{H}$). For simplicity, we can let $X=\Gamma\backslash\mathbb{H}$. Then if $k:\mathbb{H}\times\mathbb{H}$ is a point pair invariant, and $K(z,w)=\sum_{\gamma\in\Gamma}k(\gamma z,w)$ is the averaging of $k$ over $\Gamma$. Then if we define the operator $T_k:L^2(X)\rightarrow L^2(X)$ defined by $f\mapsto \int_Xk(z,w)d\mu(w)$.
Now if we take $\{u_m(z)\}$ be a complete set of eigenvectors for $T_k$ such that $T_ku_m=\lambda_k u_m$, then considering the spectral parameters $t_m$ where $\lambda_m=\frac{1}{4}+t_m^2$, then we will have that $T_k u_m=h(t_m)u_m$ and $h$ is a suitable test function.
Now we have that Selberg's pretrace formula says that $$ K(z,w)=\sum_{m=0}^\infty h(t_m)u_m(z)u_m(w) $$ Now the classical Selberg's trace formula is obtained from the above by setting $z=w$ and integrating over $X$. This gives $$ \sum_{m=0}^\infty h(t_m)=\int_X K(z,z)d\mu(z)=\int_{\Gamma\backslash\mathbb{H}}\sum_{\gamma\in\Gamma}k(\gamma z,z)d\mu(z) $$ Then breaking up the righthand side up into conjugacy classes and examining how the integrals work gives the classical version of Selberg's trace formula.
What I am interested is a version of Selberg's trace formula involving the Hecke eigenvalues (as the $\lambda_m$ are the Laplace eigenvalues). Thus, if I let $T_n$ be the Hecke operator, and if I choose my basis $u_m$ to also simultaneously be Hecke eigenfunctions such that $T_nu_m=\lambda_m(n)$ (I know notation is confusing, but I will use $\lambda_m(n)$ to be the $n$th Hecke eigenvalue associated to the function $u_m$). Now if we start off with Selberg's pretrace formula, and we apply the Hecke operator $T_n$ (in the variable $z$), let's start with the spectral side (i.e. the righthand side): $$ T_n\left(\sum_{m=0}^\infty h(t_m)u_m(z)u_m(w)\right)=\sum_{m=0}^\infty h(t_m)\lambda_m(n)u_m(z)u_m(w) $$ Then setting $z=w$, and integrating over $X$, and using the fact we choose the $u_m$ to be orthonormal, we will have that the left-hand side will become $$ \sum_{m=0}^\infty h(t_m)\lambda_m(n) $$ Now if we take the right-hand side and we apply $T_n$, then we find that $$ T_n\left(K(z,w)\right)=T_n\left(\sum_{\gamma\in\Gamma}k(\gamma z,w)\right)=\sum_{\gamma\in R(n)}k(\gamma z,w) $$ where we have $R(n)$ denote integral matrices of determinant $n$. Then if we set $z=w$ and integrate we should get the other side of the trace formula. Now, I am curious if there is a reference that goes through this side of the trace formula after applying the Hecke operators (going through the identity, elliptic, parabolic, and hyperbolic contributions). If such a reference is known as I am sure it might be a lot of details I would appreciate to see it.
