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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, $$ \sum_{t_j\in \mathbb R\atop{|t_j|<T}}1=c_T\mathrm{vol}(\Gamma_0(N)\backslash \mathbb H) +O\left(\frac{\mathrm{Vol}(\Gamma_0(N)\backslash\mathbb H)}{\log N}\right)?$$

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    $\begingroup$ I have found the answer in Page 170 in Iwaniec' book, spectral methods of automorphic forms. Thanks for the help of Prof. Guanghua Ji. $\endgroup$ – Qinghua Pi Mar 14 at 5:41
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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, 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 $0$).

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  • $\begingroup$ Thanks for your answer. $\endgroup$ – Qinghua Pi Mar 14 at 12:49

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