Bounding sup-norms of cusp forms of large level

Let f be an $L^2$-normalized weight zero Hecke-Maass cusp form of square-free level N, character $χ$ and Laplacian eigenvalue $λ\geq 1/4$. It is shown that $\| f \|_{\infty} \ll_λ N^{-1/37}$, from which the hybrid bound $\|f \|_{\infty} \ll λ^{1/4} (Nλ)^{-δ}$ (for some $δ> 0$) is derived. The first bound holds also for $f = y^{k/2}F$ where F is a holomorphic cusp form of weight k with the implied constant now depending on k.

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