{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2010:KEF3KEWDEBK6R65SATKV2CYL6P","short_pith_number":"pith:KEF3KEWD","schema_version":"1.0","canonical_sha256":"510bb512c32055e8fbb204d55d0b0bf3f3eec555d3be46a8330053e3660f4465","source":{"kind":"arxiv","id":"1008.4008","version":1},"attestation_state":"computed","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 $d_k+1=\\dim_{\\CC} M_{2k}$."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1008.4008","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-08-24T10:53:51Z","cross_cats_sorted":[],"title_canon_sha256":"ebfc3316a827c13ba65927095da77c084bf21782073c35ec2020d03072fa1d20","abstract_canon_sha256":"cb055d126136c244ad338bd894fd24e94c86796ea24b7fc62deb1fa09c3c56b1"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:41:54.350582Z","signature_b64":"+lqUhIS1As21bG2Ok+X/gR/Af204PzH0R/1IJ5w6Afai4k8MoNfeL9hanNVYPd0+g1Psae9s1WaQs2e8y3UVAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"510bb512c32055e8fbb204d55d0b0bf3f3eec555d3be46a8330053e3660f4465","last_reissued_at":"2026-05-18T04:41:54.350093Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:41:54.350093Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1008.4008","created_at":"2026-05-18T04:41:54.350166+00:00"},{"alias_kind":"arxiv_version","alias_value":"1008.4008v1","created_at":"2026-05-18T04:41:54.350166+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1008.4008","created_at":"2026-05-18T04:41:54.350166+00:00"},{"alias_kind":"pith_short_12","alias_value":"KEF3KEWDEBK6","created_at":"2026-05-18T12:26:09.077623+00:00"},{"alias_kind":"pith_short_16","alias_value":"KEF3KEWDEBK6R65S","created_at":"2026-05-18T12:26:09.077623+00:00"},{"alias_kind":"pith_short_8","alias_value":"KEF3KEWD","created_at":"2026-05-18T12:26:09.077623+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/KEF3KEWDEBK6R65SATKV2CYL6P","json":"https://pith.science/pith/KEF3KEWDEBK6R65SATKV2CYL6P.json","graph_json":"https://pith.science/api/pith-number/KEF3KEWDEBK6R65SATKV2CYL6P/graph.json","events_json":"https://pith.science/api/pith-number/KEF3KEWDEBK6R65SATKV2CYL6P/events.json","paper":"https://pith.science/paper/KEF3KEWD"},"agent_actions":{"view_html":"https://pith.science/pith/KEF3KEWDEBK6R65SATKV2CYL6P","download_json":"https://pith.science/pith/KEF3KEWDEBK6R65SATKV2CYL6P.json","view_paper":"https://pith.science/paper/KEF3KEWD","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1008.4008&json=true","fetch_graph":"https://pith.science/api/pith-number/KEF3KEWDEBK6R65SATKV2CYL6P/graph.json","fetch_events":"https://pith.science/api/pith-number/KEF3KEWDEBK6R65SATKV2CYL6P/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/KEF3KEWDEBK6R65SATKV2CYL6P/action/timestamp_anchor","attest_storage":"https://pith.science/pith/KEF3KEWDEBK6R65SATKV2CYL6P/action/storage_attestation","attest_author":"https://pith.science/pith/KEF3KEWDEBK6R65SATKV2CYL6P/action/author_attestation","sign_citation":"https://pith.science/pith/KEF3KEWDEBK6R65SATKV2CYL6P/action/citation_signature","submit_replication":"https://pith.science/pith/KEF3KEWDEBK6R65SATKV2CYL6P/action/replication_record"}},"created_at":"2026-05-18T04:41:54.350166+00:00","updated_at":"2026-05-18T04:41:54.350166+00:00"}