{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:KEF3KEWDEBK6R65SATKV2CYL6P","short_pith_number":"pith:KEF3KEWD","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"},"canonical_sha256":"510bb512c32055e8fbb204d55d0b0bf3f3eec555d3be46a8330053e3660f4465","source":{"kind":"arxiv","id":"1008.4008","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1008.4008","created_at":"2026-05-18T04:41:54Z"},{"alias_kind":"arxiv_version","alias_value":"1008.4008v1","created_at":"2026-05-18T04:41:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1008.4008","created_at":"2026-05-18T04:41:54Z"},{"alias_kind":"pith_short_12","alias_value":"KEF3KEWDEBK6","created_at":"2026-05-18T12:26:09Z"},{"alias_kind":"pith_short_16","alias_value":"KEF3KEWDEBK6R65S","created_at":"2026-05-18T12:26:09Z"},{"alias_kind":"pith_short_8","alias_value":"KEF3KEWD","created_at":"2026-05-18T12:26:09Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:KEF3KEWDEBK6R65SATKV2CYL6P","target":"record","payload":{"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"},"canonical_sha256":"510bb512c32055e8fbb204d55d0b0bf3f3eec555d3be46a8330053e3660f4465","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"},"source_kind":"arxiv","source_id":"1008.4008","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:41:54Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"snaCgprzKvM7MKFePIv9ZSClZ22WWbiIWZotFKPsjjo5Xxd6mUKKcTCf/jB7OlaelkpQrZk6EjnFlck3vMBrAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T13:37:49.625215Z"},"content_sha256":"94910b47ab98d26d5f175515a7bbf91573c69ac64ec7ad1e239b38fdfeab43f1","schema_version":"1.0","event_id":"sha256:94910b47ab98d26d5f175515a7bbf91573c69ac64ec7ad1e239b38fdfeab43f1"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:KEF3KEWDEBK6R65SATKV2CYL6P","target":"graph","payload":{"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:41:54Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Zi3ElYHdXyuemhejRUbuUNWZSaYyGC6Yyh4eEAxJK5sGF4bWfEcjLI1/KGrVwVeF8+AuCiykVvJiTjisxeDGDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T13:37:49.625578Z"},"content_sha256":"53993e3f7beb37fc5db34b32ce371d12fe9ee46ca6c6daee9537976df538d1bc","schema_version":"1.0","event_id":"sha256:53993e3f7beb37fc5db34b32ce371d12fe9ee46ca6c6daee9537976df538d1bc"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/KEF3KEWDEBK6R65SATKV2CYL6P/bundle.json","state_url":"https://pith.science/pith/KEF3KEWDEBK6R65SATKV2CYL6P/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/KEF3KEWDEBK6R65SATKV2CYL6P/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-05T13:37:49Z","links":{"resolver":"https://pith.science/pith/KEF3KEWDEBK6R65SATKV2CYL6P","bundle":"https://pith.science/pith/KEF3KEWDEBK6R65SATKV2CYL6P/bundle.json","state":"https://pith.science/pith/KEF3KEWDEBK6R65SATKV2CYL6P/state.json","well_known_bundle":"https://pith.science/.well-known/pith/KEF3KEWDEBK6R65SATKV2CYL6P/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:KEF3KEWDEBK6R65SATKV2CYL6P","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"cb055d126136c244ad338bd894fd24e94c86796ea24b7fc62deb1fa09c3c56b1","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-08-24T10:53:51Z","title_canon_sha256":"ebfc3316a827c13ba65927095da77c084bf21782073c35ec2020d03072fa1d20"},"schema_version":"1.0","source":{"id":"1008.4008","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1008.4008","created_at":"2026-05-18T04:41:54Z"},{"alias_kind":"arxiv_version","alias_value":"1008.4008v1","created_at":"2026-05-18T04:41:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1008.4008","created_at":"2026-05-18T04:41:54Z"},{"alias_kind":"pith_short_12","alias_value":"KEF3KEWDEBK6","created_at":"2026-05-18T12:26:09Z"},{"alias_kind":"pith_short_16","alias_value":"KEF3KEWDEBK6R65S","created_at":"2026-05-18T12:26:09Z"},{"alias_kind":"pith_short_8","alias_value":"KEF3KEWD","created_at":"2026-05-18T12:26:09Z"}],"graph_snapshots":[{"event_id":"sha256:53993e3f7beb37fc5db34b32ce371d12fe9ee46ca6c6daee9537976df538d1bc","target":"graph","created_at":"2026-05-18T04:41:54Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"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 $d_k+1=\\dim_{\\CC} M_{2k}$.","authors_text":"Shinji Fukuhara","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-08-24T10:53:51Z","title":"A new basis for the space of modular forms"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1008.4008","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:94910b47ab98d26d5f175515a7bbf91573c69ac64ec7ad1e239b38fdfeab43f1","target":"record","created_at":"2026-05-18T04:41:54Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"cb055d126136c244ad338bd894fd24e94c86796ea24b7fc62deb1fa09c3c56b1","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-08-24T10:53:51Z","title_canon_sha256":"ebfc3316a827c13ba65927095da77c084bf21782073c35ec2020d03072fa1d20"},"schema_version":"1.0","source":{"id":"1008.4008","kind":"arxiv","version":1}},"canonical_sha256":"510bb512c32055e8fbb204d55d0b0bf3f3eec555d3be46a8330053e3660f4465","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"510bb512c32055e8fbb204d55d0b0bf3f3eec555d3be46a8330053e3660f4465","first_computed_at":"2026-05-18T04:41:54.350093Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:41:54.350093Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"+lqUhIS1As21bG2Ok+X/gR/Af204PzH0R/1IJ5w6Afai4k8MoNfeL9hanNVYPd0+g1Psae9s1WaQs2e8y3UVAw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:41:54.350582Z","signed_message":"canonical_sha256_bytes"},"source_id":"1008.4008","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:94910b47ab98d26d5f175515a7bbf91573c69ac64ec7ad1e239b38fdfeab43f1","sha256:53993e3f7beb37fc5db34b32ce371d12fe9ee46ca6c6daee9537976df538d1bc"],"state_sha256":"53acf7830f749dabbd38cdc191b4561212f88ccc3e80345515b9ec229556dcae"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"eSC4GuvPSX0PlKWonJh9kIvKGgV7qMs3QYlxlP5jWKTKF4F+UFRhbYBmZMsdeTqmCN9l8LJJlQm96lfVQuD1Dg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-05T13:37:49.627763Z","bundle_sha256":"e5d37e423051e4d9ea24c5f8a37fbf14496660f85b256378a70d3d5922049aa6"}}