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Let $f$ be an $L^2$-normalized Hecke--Maass cuspidal newform of level $N$,character $\chi$ and Laplace eigenvalue $\lambda$. Let $N_1$ denote thesmallest integer such that $N|N_1^2$ and $N_0$ denote the largest integer suchthat $N_0^2 |N$. 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 theN-aspect; it had been shown by the author that $|f|_\infty \ll_{\lambda,\epsilon} N^{5/12+\epsilon}$ provided $M=1$. The present result significantlyimproves 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-adicrepresentation theoretic techniques and in particular a detailed study ofWhittaker newforms and matrix coefficients for $GL_2(F)$ where $F$ is a localfield.

Hybrid sup-norm bounds for Maass newforms of powerful level