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](http://www.sci.ccny.cuny.edu/%7Ebfeigon/Average_L-values.pdf) seems relevant), but I am unqualified to determine what is known for the sum I wrote above.



Bounding a Sum of Adjoint L-Function Values
Here is a naive way I think you can get some bounds. Warning: I have not thought through this too carefully, and the bounds from such an approach might not be as good as you want in any case. 


The adjoint $L$-value is essentially the Petersson norm $(f,f)$. If $f$ is a new form, then it is $\frac{2^{k}}{N} (f,f)$ with normalization as in that Feigon-Whitehouse paper. So you want to bound $(f,f)$. Well, the Petersson norm is equivalent to the sup norm (of $y^{k/2}|f|$) on $S(k,N)$. Asymptotic bounds for the sup norm have been studied analytically, for instance by [Blomer-Holowinsky](http://arxiv.org/abs/0808.1525) (which also contains a summary of the relation between the sup norm and Petersson norm) in the case of square free level and recently by [Saha](http://www.maths.bris.ac.uk/~as12313/research/sup-norm-level.pdf) in greater generality. This gives you bounds for individual newforms, and a slightly cruder bound for a general cusp form. Then you can use the dimension formula to at least get a rough bound on this sum. One can maybe do better by trying to filter forms in $S(k,N)$ according to their "true level."


