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By the Hasse bound, $a_E(p) = 2\\sqrt{p} \\cos \\theta_p$ for a unique $\\theta_p \\in [0, \\pi]$. In this paper, we prove that the least prime $p$ such that $\\theta_p \\in [\\alpha, \\beta] \\subset [0, \\pi]$ satisfies \\[ p \\ll \\left(\\frac{N_E}{\\beta - \\alpha}\\right)^A, \\] where $N_E$ is the conductor of $E$ and the implied constant and exponent $A > 2$ are absolute and effectively computable. Our result is an analogu"},"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":"1506.09170","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-06-30T17:37:02Z","cross_cats_sorted":[],"title_canon_sha256":"1626f1be16ea2bbd4cff3fd6ab6daccf02ffbdde3e4ddd067ccc4eb98c4a4588","abstract_canon_sha256":"d57345cc8388a18de45a8c72612970e063998e76a19875f1ae63f14da995fbb6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:09:07.008352Z","signature_b64":"oy8wyU8kwTNlog9zCFKZSr9+CiWO3OlHWawY9u6kwgGCpDWqTC80pOKDrv7GpJL8rJtDR2Higt/jkZT6CGyNCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c35297df45de1452fed96d75e87dc14fa1dd1b99e4c08b0d5e493efd2a4d6dd1","last_reissued_at":"2026-05-18T01:09:07.007751Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:09:07.007751Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Linnik's Theorem for Sato-Tate Laws on Elliptic Curves with Complex Multiplication","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Ashvin Swaminathan, Evan Chen, Peter S. Park","submitted_at":"2015-06-30T17:37:02Z","abstract_excerpt":"Let $E/\\mathbb{Q}$ be an elliptic curve with complex multiplication (CM), and for each prime $p$ of good reduction, let $a_E(p) = p + 1 - \\#E(\\mathbb{F}_p)$ denote the trace of Frobenius. By the Hasse bound, $a_E(p) = 2\\sqrt{p} \\cos \\theta_p$ for a unique $\\theta_p \\in [0, \\pi]$. In this paper, we prove that the least prime $p$ such that $\\theta_p \\in [\\alpha, \\beta] \\subset [0, \\pi]$ satisfies \\[ p \\ll \\left(\\frac{N_E}{\\beta - \\alpha}\\right)^A, \\] where $N_E$ is the conductor of $E$ and the implied constant and exponent $A > 2$ are absolute and effectively computable. 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