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 $N|N_1^2$ and $N_0$ denote the largest integer such that $N_0^2 |N$. Let $M$ denote the conductor of $\chi$ and define $M_1= M/\gcd(M,N_1)$. In this paper, we prove the bound $|f|_\infty$ $\ll_{\epsilon}$ $N_0^{1/6 + \epsilon} N_1^{1/3+\epsilon} M_1^{1/2} \lambda^{5/24+\epsilon}$, which generalizes and strengthens previously known upper bounds for $|f|_\infty$. This is the first time a hybrid bound (i.e., involving both $N$ and $\lambda$) has been established for $|f|_\infty$ 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 $GL_2(F)$ where $F$ is a local field.