# Bounding a Sum of Adjoint L-Function Values

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

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

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 (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 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."