The Weyl law for Maass cusp forms for $SL_2(\mathbb Z)$ was obtained by Selberg firstly and has been generalized to various caces (Duistermaat-Guillemin, Lapid-Müller and others). For example, one of the version of Weyl's law for maass wave forms for congrunce subgroup is $$ \sum_{t_j\in \mathbb R\atop{|t_j|<T}}1=\frac{\mathrm{Vol}(\Gamma_0(N)\backslash \mathbb{H})}{2\pi}T^2+O\left(T\log T\right). $$ Here $\{\pm t_j\}$ is the set of spectral parameters of Maass form and $\lambda_j=\frac{1}{4}+t_j^2$ is the eigenvalue of the Laplacian operator.

For $N$ large, the Selberg eigenvalue conjecture has not been proved. Some result (Xianjin Li) indicated that the multiplicity of exceptional eigenvalue (if it exists) can be arbituary large and is bounded by a constant depending on $N$.

Is it possible to obtain an upper bound of the exceptional eigenvalues such as $$ \sum_{t_j\in i\mathbb R\atop{0<|t_j|<1/2}}1\ll \frac{{\mathrm{Vol}(\Gamma_0(N)\backslash\mathbb H)}}{\log N},\quad N\rightarrow\infty $$ or equivalently, can we get rid of the contribution of exceptional eigenvalues in the proof of Weyl's law to obtain a version in level aspect, $$ \sum_{t_j\in \mathbb R\atop{|t_j|<T}}1=c_T\mathrm{vol}(\Gamma_0(N)\backslash \mathbb H) +O\left(\frac{\mathrm{Vol}(\Gamma_0(N)\backslash\mathbb H)}{\log N}\right)?$$