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We say that $f$ is non-ordinary at a prime $p$ if there is a prime ideal $\\mathfrak{p}\\subset O_L$ above $p$ for which $a_f(p)\\equiv 0 \\ (mod\\ {\\mathfrak{p}})$. For any finite set of primes $S$, we prove that there are normalized Hecke eigenforms which are non-ordinary for each $p\\in S$. The proof is elementary and follows from a generalization of work of Choie, Kohne"},"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":"1511.06725","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-11-20T19:04:52Z","cross_cats_sorted":[],"title_canon_sha256":"33dacd286ec068bd13590fe1d92820898d027b6f5a2a3c2a66d9a3942f541a5e","abstract_canon_sha256":"67f630fecd5c9f54b2aef0bfd93f49eb0f9b79b5b0d25410cfe7d7dc426e17c3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:23:15.888519Z","signature_b64":"2ybkzuPdrVn5EE9NSjdNw2aWLvW21tIsfrZo31eB2lGoDvZRcQQUhbVE4zn0PWUIfLtYvIG3tMfv+qG+brcuDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"db5569961a6c69351be4f69fb465ef258f6d5a7994ac83435d8978ecfb2ec8e6","last_reissued_at":"2026-05-18T01:23:15.887836Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:23:15.887836Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A Note On Non-ordinary Primes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Ken Ono, Seokho Jin, Wenjun Ma","submitted_at":"2015-11-20T19:04:52Z","abstract_excerpt":"Suppose that $O_L$ is the ring of integers of a number field $L$, and suppose that $f(z)=\\sum_{n=1}^\\infty a_f(n)q^n\\in S_k\\cap O_L[[q]]$ (note: $q := e^{2\\pi iz}$) is a normalized Hecke eigenform for $\\mathrm{SL}_2(\\mathbb{Z})$. We say that $f$ is non-ordinary at a prime $p$ if there is a prime ideal $\\mathfrak{p}\\subset O_L$ above $p$ for which $a_f(p)\\equiv 0 \\ (mod\\ {\\mathfrak{p}})$. For any finite set of primes $S$, we prove that there are normalized Hecke eigenforms which are non-ordinary for each $p\\in S$. 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