Hybrid sup-norm bounds for Maass newforms of powerful level

Let $f$ be an $L^2$-normalized Hecke–Maass cuspidal newform of level $N$, character $\chi$ and Laplace eigenvalue $\lambda$. Let $N_1$ denote the smallest integer such that $N\mathrm{\mid }{N}_1^2$ and $N_0$ denote the largest integer such that $N_0^2\mathrm{\mid }N$. Let $M$ denote the conductor of $\chi$ and define $M_1= M/\gcd(M,N_1)$\mathrm{.}Inthispaper,weprovethebound$|f|\infty$ ${\ll }_{ϵ}$ $N_0^{1\mathrm{/}6+ϵ}{N}_1^{1\mathrm{/}3+ϵ}{M}_1^{1\mathrm{/}2}{\lambda }^{5\mathrm{/}24+ϵ}$, which generalizes and strengthens previously known upper bounds for $\mathrm{\mid }f{\mathrm{\mid }}_{\mathrm{\infty }}$. This is the first time a hybrid bound (i.e., involving both $N$ and $\lambda$) has been established for $\mathrm{\mid }f{\mathrm{\mid }}_{\mathrm{\infty }}$ in the case of non-squarefree $N$. The only previously known bound in the non-squarefree case was in the N-aspect; it had been shown by the author that $|f|\infty \ll_{\lambda, \epsilon} N^{5/12+\epsilon}$provided$M=1$. The present result significantly improves the exponent of $N$ in the above case. If $N$ is a squarefree integer, our bound reduces to $|f|\infty \ll\epsilon N^{1/3 + \epsilon}\lambda^{5/24 + \epsilon}$, which was previously proved by Templier. The key new feature of the present work is a systematic use of p-adic representation theoretic techniques and in particular a detailed study of Whittaker newforms and matrix coefficients for $G{L}_2(F)$ where $F$ is a local field.