{"paper":{"title":"A new basis for the space of modular forms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Shinji Fukuhara","submitted_at":"2010-08-24T10:53:51Z","abstract_excerpt":"Let $G_{2n}$ be the Eisenstein series of weight $2n$ for the full modular group $\\Gamma=SL_2(\\ZZ)$. It is well-known that the space $M_{2k}$ of modular forms of weight $2k$ on $\\Gamma$ has a basis $\\{G_{4}^\\alpha G_{6}^\\beta\\ |\\ \\alpha,\\beta\\in\\ZZ,\\ \\alpha,\\beta\\geq 0,\\\n  4\\alpha+6\\beta=2k\\}$. In this paper we will exhibit another (simpler) basis for $M_{2k}$. It is given by\n  $\\{G_{2k}\\}\\cup\\{G_{4i}G_{2k-4i}\\ |\\ i=1,2,\\ldots,d_k\\}$ if $2k\\equiv 0\\pmod 4$, and\n  $\\{G_{2k}\\}\\cup\\{G_{4i+2}G_{2k-4i-2}\\ |\\ i=1,2,\\ldots,d_k\\}$ if $2k\\equiv 2\\pmod 4$ where $d_k+1=\\dim_{\\CC} M_{2k}$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1008.4008","kind":"arxiv","version":1},"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"}