{"paper":{"title":"Bounds for coefficients of cusp forms and extremal lattices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Jeremy Rouse, Paul Jenkins","submitted_at":"2010-12-29T17:53:57Z","abstract_excerpt":"A cusp form $f(z)$ of weight $k$ for $\\SL_{2}(\\Z)$ is determined uniquely by its first $\\ell := \\dim S_{k}$ Fourier coefficients. We derive an explicit bound on the $n$th coefficient of $f$ in terms of its first $\\ell$ coefficients. We use this result to study the non-negativity of the coefficients of the unique modular form of weight $k$ with Fourier expansion \\[F_{k,0}(z) = 1 + O(q^{\\ell + 1}).\\] In particular, we show that $k = 81632$ is the largest weight for which all the coefficients of $F_{0,k}(z)$ are non-negative. This result has applications to the theory of extremal lattices."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1012.5991","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}