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Let $M$ denote the conductor of $\\chi$ and define $M_1= M/\\gcd(M,N_1)$. In this paper, we prove the bound $|f|_\\infty$ $\\ll_{\\epsilon}$ $N_0^{1/6 + \\epsilon} N_1^{1/3+\\epsilon} M_1^{1/2} \\lambda^{5/24+\\epsilon}$, which generalizes and strengthens previously known upper bounds for $|f|_\\infty$.\n  This is the first time a hybrid bound (i.e., involving both "},"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":"1509.07489","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-09-24T19:47:16Z","cross_cats_sorted":["math.RT","math.SP"],"title_canon_sha256":"859e50be561a7e3efad513f367c78d38152979f1ac7abec4866d44e07a6622f1","abstract_canon_sha256":"e47972d68b6c84c6984b28a8975d2ca213800eb625d897f9d57bab63aa059c75"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:22:30.004038Z","signature_b64":"AjqwpyVHsf2MLRl9iHW1paZkFI8RRjPlyNIZIUgj38C7X4Gw53iFBHcr2OOMD6RAN5i5g0LpcTvLbonCKk3tBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"57b722468ad061b05bd0d43a88292ae9a68e81448c9b363da7dfb7865741d50f","last_reissued_at":"2026-05-18T00:22:30.003287Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:22:30.003287Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Hybrid sup-norm bounds for Maass newforms of powerful level","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RT","math.SP"],"primary_cat":"math.NT","authors_text":"Abhishek Saha","submitted_at":"2015-09-24T19:47:16Z","abstract_excerpt":"Let $f$ be an $L^2$-normalized Hecke--Maass cuspidal newform of level $N$, character $\\chi$ and Laplace eigenvalue $\\lambda$. 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In this paper, we prove the bound $|f|_\\infty$ $\\ll_{\\epsilon}$ $N_0^{1/6 + \\epsilon} N_1^{1/3+\\epsilon} M_1^{1/2} \\lambda^{5/24+\\epsilon}$, which generalizes and strengthens previously known upper bounds for $|f|_\\infty$.\n  This is the first time a hybrid bound (i.e., involving both "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.07489","kind":"arxiv","version":4},"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":"1509.07489","created_at":"2026-05-18T00:22:30.003419+00:00"},{"alias_kind":"arxiv_version","alias_value":"1509.07489v4","created_at":"2026-05-18T00:22:30.003419+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.07489","created_at":"2026-05-18T00:22:30.003419+00:00"},{"alias_kind":"pith_short_12","alias_value":"K63SERUK2BQ3","created_at":"2026-05-18T12:29:27.538025+00:00"},{"alias_kind":"pith_short_16","alias_value":"K63SERUK2BQ3AW6Q","created_at":"2026-05-18T12:29:27.538025+00:00"},{"alias_kind":"pith_short_8","alias_value":"K63SERUK","created_at":"2026-05-18T12:29:27.538025+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/K63SERUK2BQ3AW6Q2Q5IQKJK5G","json":"https://pith.science/pith/K63SERUK2BQ3AW6Q2Q5IQKJK5G.json","graph_json":"https://pith.science/api/pith-number/K63SERUK2BQ3AW6Q2Q5IQKJK5G/graph.json","events_json":"https://pith.science/api/pith-number/K63SERUK2BQ3AW6Q2Q5IQKJK5G/events.json","paper":"https://pith.science/paper/K63SERUK"},"agent_actions":{"view_html":"https://pith.science/pith/K63SERUK2BQ3AW6Q2Q5IQKJK5G","download_json":"https://pith.science/pith/K63SERUK2BQ3AW6Q2Q5IQKJK5G.json","view_paper":"https://pith.science/paper/K63SERUK","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1509.07489&json=true","fetch_graph":"https://pith.science/api/pith-number/K63SERUK2BQ3AW6Q2Q5IQKJK5G/graph.json","fetch_events":"https://pith.science/api/pith-number/K63SERUK2BQ3AW6Q2Q5IQKJK5G/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/K63SERUK2BQ3AW6Q2Q5IQKJK5G/action/timestamp_anchor","attest_storage":"https://pith.science/pith/K63SERUK2BQ3AW6Q2Q5IQKJK5G/action/storage_attestation","attest_author":"https://pith.science/pith/K63SERUK2BQ3AW6Q2Q5IQKJK5G/action/author_attestation","sign_citation":"https://pith.science/pith/K63SERUK2BQ3AW6Q2Q5IQKJK5G/action/citation_signature","submit_replication":"https://pith.science/pith/K63SERUK2BQ3AW6Q2Q5IQKJK5G/action/replication_record"}},"created_at":"2026-05-18T00:22:30.003419+00:00","updated_at":"2026-05-18T00:22:30.003419+00:00"}