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,
$$
\sum\_{t\_j\in \mathbb R\atop{|t\_j|<T}}1=c\_T\mathrm{vol}(\Gamma\_0(N)\backslash \mathbb H) +O\left(\frac{\mathrm{Vol}(\Gamma\_0(N)\backslash\mathbb H)}{\log N}\right)?$$



The exceptional eigenvalues and Weyl's law in level aspect
You are asking about density theorems for exceptional eigenvalues. As you mention, this has been answered by Iwaniec: if $\mathcal{B}\_0(q)$ denotes an orthonormal basis of Maass forms of weight $0$, level $q$, and trivial nebentypus with spectral parameter $t\_f \in [0,\infty)$ or $it\_f \in (0,1/2)$ (the latter the exceptional eigenvalues), Iwaniec gives the bound
\[\#\{f \in \mathcal{B}\_0(q) : it\_f \geq \alpha\} \ll\_{\varepsilon} \mathrm{vol}(\Gamma\_0(q) \backslash \mathbb{H})^{1 - 4\alpha + \varepsilon}.\]
This implies the Selberg eigenvalue bound $\lambda\_f = 1/4 + t\_f^2 \geq 3/16$.


This has recently been improved in my [paper](https://doi.org/10.2140/ant.2018.12.1581), where I look at density theorems for exceptional Laplacian and Hecke eigenvalues simultaneously for congruence subgroups $\Gamma\_0(q)$, $\Gamma\_1(q)$, and $\Gamma(q)$.


Note that density theorems of this form (with the presence of $\varepsilon$) are not good when the exceptional eigenvalue is extremely close to $1/4$ (or equivalently the exceptional spectral parameter is extremely close to $0$).


