On sup-norms of cusp forms of powerful level (Q1676093)

Summary: Let \(f\) be an \(L^2\)-normalized Hecke-Maass cuspidal newform of level \(N\) and Laplace eigenvalue \(\lambda\). It is shown that \(\|f\|_\infty\ll_{\lambda,\varepsilon} N^{-1/12+\varepsilon}\) for any \(\varepsilon>0\). The exponent is further improved in the case when \(N\) is not divisible by ``small squares''. Our work extends and generalizes previously known results in the special case of \(N\) squarefree.