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We study the twisted first moment of the triple product $L$-function $L(\\frac{1}{2}, \\pi \\otimes \\pi_1 \\otimes \\pi_2)$ and the Hecke eigenvalues $\\lambda_\\pi (\\mathfrak{l})$, where $\\pi$ is a unitary automorphic representation of $\\mathrm{PGL}_2(\\mathbb{A}_F)$ and $\\mathfrak{l}$ is an integral ideal coprimes with the finite analytic conductor $C(\\pi \\otimes \\pi_1 \\otimes \\pi_2)$. The estimation becomes a r"},"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":"2501.04022","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2024-12-27T18:59:24Z","cross_cats_sorted":[],"title_canon_sha256":"a8525ae0d9b0a2302ea019b50cc03b3fcf609fc1dbb0cfdc34e259c3d2fc3c40","abstract_canon_sha256":"7901d6e527a0840b4c1ac9f16582978bbd5124f37df83b6207878200aafb2729"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:58:28.700118Z","signature_b64":"VayKiHLmH5TCrV1Vjm5x5iw1uY2Y4kNwl2UJj+uZyY5c1Inov5MWC/QIS8hzXus8DBj2StiTlszIIAASZaWiBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"65db88dc113721dfb8fad4b0ed4e508e674e9a75181a8c15d12b3207dd5b715e","last_reissued_at":"2026-07-05T09:58:28.699600Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:58:28.699600Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Spectral Reciprocity and Hybrid Subconvexity Bound for triple product $L$-functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Xinchen Miao","submitted_at":"2024-12-27T18:59:24Z","abstract_excerpt":"Let $F$ be a number field with adele ring $\\mathbb{A}_F$, $\\pi_1, \\pi_2$ be two unitary cuspidal automorphic representations of $\\mathrm{PGL}_2(\\mathbb{A}_F)$ with finite analytic conductor. 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