Fix integers $k\geq2$ and $N>1$, and let $S(k,N)$ denote the normalized new Hecke eigenforms in $S_k(\Gamma_1(N))$. [If it makes my question easier to answer, feel free to replace this with $\Gamma_0(N)$].

For $f \in S(k,N)$, one can define an associated adjoint L-function $L(s,\mathrm{ad}\ f)$. Of particular interest is the special value at $s=1$, which is related to congruences between modular forms. There is a period $\Omega_f$ such that $\frac{L(1,\mathrm{ad}\ f)}{\Omega_f}$ is algebraic.

I am interested in the finite sum

$\displaystyle\sum_{f \in S(k,N)}\frac{L(1,\mathrm{ad}\ f)}{\Omega_f} $.

What, if anything, can be said about this sum? Can we bound it?

There appears to be a vast amount of literature on related topics (e.g. the paper http://www.sci.ccny.cuny.edu/~bfeigon/Average_L-values.pdf seems relevant), but I am unqualified to determine what is known for the sum I wrote above.