  We obtain an upper bound for the dimension of the cuspidal automorphic forms
for $\mathrm{GL}_2$ over a number field, whose archimedean local
representations are not tempered. More precisely, we prove the following
result.
  Let $F$ be a number field and $\mathbb{A}_{F}$ be the ring of adeles of $F$.
Let $\mathcal{O}_{F}$ be the ring of integers of $F$. Let
$\mathfrak{X}_{F,\mathrm{ex}}$ be the set of irreducible cuspidal automorphic
representations $\pi$ of $\mathrm{GL}_2(\mathbb{A}_{F})$ with the trivial
central character such that for each archimedean place $v$ of $F$, the local
representation of $\pi$ at $v$ is an unramified principal series and is not
tempered. For an ideal $J$ of $\mathcal{O}_{F}$, let $\mathrm{K}_{0}(J)$ be the
subgroup of $\mathrm{GL}_2(\mathbb{A}_{F})$ corresponding to $\Gamma_0(J)
\subset \mathrm{SL}_2(\mathcal{O}_F)$. Let $r_1$ be the number of real
embeddings of $F$ and $r_2$ be the number of conjugate pairs of complex
embeddings of $F$. Using the Arthur-Selberg trace formula, we have
\begin{equation*}
  \sum_{\pi\in \mathfrak{X}_{F,\mathrm{ex}}} \dim \pi^{\mathrm{K}_0(J)}
  \ll_{F} \frac{[\mathrm{SL}_2(\mathcal{O}_{F}) : \Gamma_0(J)]}{(\log
(N_{F/\mathbb{Q}}(J)))^{2r_1+3r_2}} \quad \text{ as } \quad
|N_{F/\mathbb{Q}}(J)|\to \infty.
  \end{equation*} From this result, we obtain the result on an upper bound for
the number of Hecke-Maass cusp forms of weight $0$ on $\Gamma_0(N)$ which do
not satisfy the Selberg eigenvalue conjecture.
