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In addition, we prove that the natural density of such primes $p$ ($p\\nmid N$ and $|\\alpha_{p}|=|\\beta_{p}| = 1$) is at least $34/35$."},"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":"1405.4937","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-05-20T03:08:43Z","cross_cats_sorted":[],"title_canon_sha256":"aa596d9baf26bee6f927b131638f64017b34517137ace345d0d7523dc8d0b75e","abstract_canon_sha256":"f14e953fb4ed3179a280979593908ea7b1790af7c9c3066da32722d07299d972"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:49:31.393138Z","signature_b64":"8tWkZlSZR9hS2g0HFj3uB78BGl06p0WigUbgtzIb5LBqP41/Ju0LYveP9R88E4B1UGwthHaO/y6uw9OdK1/OBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4d1d8cebbf5b7b9ade1391ec34c89cfbd4f41abe1b32bdcd1210303e7046ac8e","last_reissued_at":"2026-05-18T02:49:31.392709Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:49:31.392709Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the Hecke Eigenvalues of Maass Forms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Fan Zhou, Wenzhi Luo","submitted_at":"2014-05-20T03:08:43Z","abstract_excerpt":"Let $\\phi$ denote a primitive Hecke-Maass cusp form for $\\Gamma_o(N)$ with the Laplacian eigenvalue $\\lambda_\\phi=1/4+t_{\\phi}^2$. 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