On sup-norms of cusp forms of powerful level

Abstract

Let $f$ be an $L^2$-normalized Hecke–Maass cuspidal newform of level $N$ and Laplace eigenvalue $\lambda$. It is shown that $\Vert f{\Vert }_{\infty }{\ll }_{\lambda ,ϵ}{N}^{-1/12+ϵ}$ for any $ϵ>0$. The exponent is further improved in the case when $N$ is not divisible by "small squares". Our work extends and generalizes previously known results in the special case of $N$ squarefree.