Sup-norm and nodal domains of dihedral Maass forms

In this paper, we improve the sup-norm bound and the lower bound of the number of nodal domains for dihedral Maass forms, which are a distinguished sequence of Laplacian eigenfunctions on an arithmetic hyperbolic surface. More specifically, let $\varphi$ be a dihedral Maass form with spectral parameter $t_{\varphi }$, then we prove that $|\phi|\infty \ll t\phi^{3/8+\varepsilon} |\phi|2$,whichisanimprovementoverthebound$t\phi^{5/12+\varepsilon} |\phi|_2$ given by Iwaniec and Sarnak. As a consequence, we get a better lower bound for the number of nodal domains intersecting a fixed geodesic segment under the Lindel"{o}f Hypothesis. Unconditionally, we prove that the number of nodal domains grows faster than $t_{\varphi }^{1\mathrm{/}8-\epsilon }$ for any $\epsilon >0$ for almost all dihedral Maass forms.