{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:PY4X2OL5WDKYCGJI2VW73EICOH","short_pith_number":"pith:PY4X2OL5","canonical_record":{"source":{"id":"2411.14990","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2024-11-22T15:00:41Z","cross_cats_sorted":[],"title_canon_sha256":"92825b1a5a3cdcc59656d5aef38422c9cf4373549974614e7aedeceef6b330db","abstract_canon_sha256":"70ac3094e6ce691bd4ee7e8e81f41b27b26a8d5dc6b2f129ede91364b5751e92"},"schema_version":"1.0"},"canonical_sha256":"7e397d397db0d5811928d56dfd910271c80867b02194755153b07101b5cfe0ad","source":{"kind":"arxiv","id":"2411.14990","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2411.14990","created_at":"2026-07-05T09:39:14Z"},{"alias_kind":"arxiv_version","alias_value":"2411.14990v1","created_at":"2026-07-05T09:39:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2411.14990","created_at":"2026-07-05T09:39:14Z"},{"alias_kind":"pith_short_12","alias_value":"PY4X2OL5WDKY","created_at":"2026-07-05T09:39:14Z"},{"alias_kind":"pith_short_16","alias_value":"PY4X2OL5WDKYCGJI","created_at":"2026-07-05T09:39:14Z"},{"alias_kind":"pith_short_8","alias_value":"PY4X2OL5","created_at":"2026-07-05T09:39:14Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:PY4X2OL5WDKYCGJI2VW73EICOH","target":"record","payload":{"canonical_record":{"source":{"id":"2411.14990","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2024-11-22T15:00:41Z","cross_cats_sorted":[],"title_canon_sha256":"92825b1a5a3cdcc59656d5aef38422c9cf4373549974614e7aedeceef6b330db","abstract_canon_sha256":"70ac3094e6ce691bd4ee7e8e81f41b27b26a8d5dc6b2f129ede91364b5751e92"},"schema_version":"1.0"},"canonical_sha256":"7e397d397db0d5811928d56dfd910271c80867b02194755153b07101b5cfe0ad","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:39:14.651280Z","signature_b64":"yJhr/OdALcxMPeU7i3Mx+JdkfrZjrhHCkt9zwq7g40KZuUanE+hF3M5Wm13Uz0DDeNAve4PWwkr+3psh03hXAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7e397d397db0d5811928d56dfd910271c80867b02194755153b07101b5cfe0ad","last_reissued_at":"2026-07-05T09:39:14.650888Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:39:14.650888Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2411.14990","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":"2026-07-05T09:39:14Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"qIec1QLW7JdRaEp+Ua/D4crwUScfYGwRpZoOXp+KHLyo/vmSxeivTXpJTTWjb9Gnw5DGvCWAerW37BN8yUH9AA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-12T23:46:44.605203Z"},"content_sha256":"ad89d2ebfc99c8be01652380d327ab4beb4e72841e96b136f913601fec8564c8","schema_version":"1.0","event_id":"sha256:ad89d2ebfc99c8be01652380d327ab4beb4e72841e96b136f913601fec8564c8"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:PY4X2OL5WDKYCGJI2VW73EICOH","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the vanishing of coefficients of $\\eta^{26}","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"S. Krishnamoorthy, T. Dalal","submitted_at":"2024-11-22T15:00:41Z","abstract_excerpt":"J.-P. Serre, in his paper [1], established a sufficient condition on $n$ for the $n$-th coefficient of the series $\\eta^{26}$ to vanish. However, the question that whether this is a necessary condition remained unanswered. In this paper, using the theory of Hecke eigenforms explored by Serre, we prove some partial cases for the converse part."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2411.14990","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2411.14990/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-05T09:39:14Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"nmgu/jS/owZGncagU/sHey2+YKOCr/17HK3U+6ypsdOZkFy/gV59O8rUCt1Bc6lRvhdk5b6W+KmCxmfPt2P4DQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-12T23:46:44.605876Z"},"content_sha256":"8524dcb16514ea4f8c8f4ab2992f1ab00e0287005a21f8ced6acd547d8d12a89","schema_version":"1.0","event_id":"sha256:8524dcb16514ea4f8c8f4ab2992f1ab00e0287005a21f8ced6acd547d8d12a89"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/PY4X2OL5WDKYCGJI2VW73EICOH/bundle.json","state_url":"https://pith.science/pith/PY4X2OL5WDKYCGJI2VW73EICOH/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/PY4X2OL5WDKYCGJI2VW73EICOH/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":"2026-08-12T23:46:44Z","links":{"resolver":"https://pith.science/pith/PY4X2OL5WDKYCGJI2VW73EICOH","bundle":"https://pith.science/pith/PY4X2OL5WDKYCGJI2VW73EICOH/bundle.json","state":"https://pith.science/pith/PY4X2OL5WDKYCGJI2VW73EICOH/state.json","well_known_bundle":"https://pith.science/.well-known/pith/PY4X2OL5WDKYCGJI2VW73EICOH/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:PY4X2OL5WDKYCGJI2VW73EICOH","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":"70ac3094e6ce691bd4ee7e8e81f41b27b26a8d5dc6b2f129ede91364b5751e92","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2024-11-22T15:00:41Z","title_canon_sha256":"92825b1a5a3cdcc59656d5aef38422c9cf4373549974614e7aedeceef6b330db"},"schema_version":"1.0","source":{"id":"2411.14990","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2411.14990","created_at":"2026-07-05T09:39:14Z"},{"alias_kind":"arxiv_version","alias_value":"2411.14990v1","created_at":"2026-07-05T09:39:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2411.14990","created_at":"2026-07-05T09:39:14Z"},{"alias_kind":"pith_short_12","alias_value":"PY4X2OL5WDKY","created_at":"2026-07-05T09:39:14Z"},{"alias_kind":"pith_short_16","alias_value":"PY4X2OL5WDKYCGJI","created_at":"2026-07-05T09:39:14Z"},{"alias_kind":"pith_short_8","alias_value":"PY4X2OL5","created_at":"2026-07-05T09:39:14Z"}],"graph_snapshots":[{"event_id":"sha256:8524dcb16514ea4f8c8f4ab2992f1ab00e0287005a21f8ced6acd547d8d12a89","target":"graph","created_at":"2026-07-05T09:39:14Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2411.14990/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"J.-P. Serre, in his paper [1], established a sufficient condition on $n$ for the $n$-th coefficient of the series $\\eta^{26}$ to vanish. However, the question that whether this is a necessary condition remained unanswered. In this paper, using the theory of Hecke eigenforms explored by Serre, we prove some partial cases for the converse part.","authors_text":"S. Krishnamoorthy, T. Dalal","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2024-11-22T15:00:41Z","title":"On the vanishing of coefficients of $\\eta^{26}"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2411.14990","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:ad89d2ebfc99c8be01652380d327ab4beb4e72841e96b136f913601fec8564c8","target":"record","created_at":"2026-07-05T09:39:14Z","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":"70ac3094e6ce691bd4ee7e8e81f41b27b26a8d5dc6b2f129ede91364b5751e92","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2024-11-22T15:00:41Z","title_canon_sha256":"92825b1a5a3cdcc59656d5aef38422c9cf4373549974614e7aedeceef6b330db"},"schema_version":"1.0","source":{"id":"2411.14990","kind":"arxiv","version":1}},"canonical_sha256":"7e397d397db0d5811928d56dfd910271c80867b02194755153b07101b5cfe0ad","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7e397d397db0d5811928d56dfd910271c80867b02194755153b07101b5cfe0ad","first_computed_at":"2026-07-05T09:39:14.650888Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T09:39:14.650888Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"yJhr/OdALcxMPeU7i3Mx+JdkfrZjrhHCkt9zwq7g40KZuUanE+hF3M5Wm13Uz0DDeNAve4PWwkr+3psh03hXAQ==","signature_status":"signed_v1","signed_at":"2026-07-05T09:39:14.651280Z","signed_message":"canonical_sha256_bytes"},"source_id":"2411.14990","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ad89d2ebfc99c8be01652380d327ab4beb4e72841e96b136f913601fec8564c8","sha256:8524dcb16514ea4f8c8f4ab2992f1ab00e0287005a21f8ced6acd547d8d12a89"],"state_sha256":"a8c7744439885e59c58dd8cef3fef2f634ecd324edbab60127401cb5d7081287"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"p8jvVr/TM+m5JJwLkztU9vk0RZjsUigsoI5HFrNo1mBy3+zQcOuEz0DJommSfmoytXMmg5dCuo3cFhq8lRQOCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-12T23:46:44.611194Z","bundle_sha256":"48185b7588cb27005ad44a72bd3ca26ace995cda9d49fc7efdf944be811c8311"}}