The sup-norm problem for newforms of large level on $\mathrm{PGL}(n)$

Let $N$ be a prime and $\phi$ be a Hecke-Maass cuspidal newform for the Hecke congruence subgroup ${\mathrm{\Gamma }}_0(N)$ in ${\mathrm{SL}}_n(\mathbb{R})$. Let $\mathrm{\Omega }$ be an adelic compactum and let ${\mathrm{\Omega }}_N$ be its projection to ${\mathrm{\Gamma }}_0(N)\mathrm{\backslash }{\mathrm{SL}}_n(\mathbb{R})\mathrm{/}\mathrm{SO}(n)$. For any prime $n$, we prove sub-baseline bounds for the sup-norm of $\phi$ restricted to ${\mathrm{\Omega }}_N$. Conditionally on GRH, we generalise this result to all $n\ge 2$. The methods involve a new reduction theory with level structure, based on generalisations of Atkin-Lehner operators.