  Let $N$ be a prime and $\phi$ be a Hecke-Maass cuspidal newform for the Hecke
congruence subgroup $\Gamma_0(N)$ in $\operatorname{SL}_n(\mathbb{R})$. Let
$\Omega$ be an adelic compactum and let $\Omega_N$ be its projection to
$\Gamma_0(N) \backslash \operatorname{SL}_n(\mathbb{R}) /
\operatorname{SO}(n)$. For any prime $n$, we prove sub-baseline bounds for the
sup-norm of $\phi$ restricted to $\Omega_N$. Conditionally on GRH, we
generalise this result to all $n \geq 2$. The methods involve a new reduction
theory with level structure, based on generalisations of Atkin-Lehner
operators.
