On sup-norms of cusp forms of powerful level. (English) Zbl 1429.11089
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.
MSC:
| 11F37 | Forms of half-integer weight; nonholomorphic modular forms |
| 11F30 | Fourier coefficients of automorphic forms |
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