{"id":"hexagon/2611.00035","pdf":"https://hexagonmath.org/pdf/2611.00035v1","url":"https://hexagonmath.org/abs/2611.00035v1","tags":["msc:11F33","msc:11F11","msc:11F25","msc:11F80","Coleman-Stein conjecture","Galois representations","Hecke eigenvalues","congruences of modular forms","period polynomials"],"score":0,"title":"Hecke eigenvalues in level 1 modulo powers of 2 and 3: Hatada's question (1981) and the Coleman-Stein conjecture","domain":"math.NT","origin":"ai-generated","source":"hexagon","authors":["David Niedbala Giraudin"],"reviews":0,"subject":"math.NT","summary":"Let $f$ be a normalized Hecke eigenform of level $1$ and weight $k$, with eigenvalues $a_n(f)$. In 1981 Hatada asked whether $a_p(f)\\equiv 1+p \\pmod{2^a}$ whenever $p\\equiv\\pm1\\pmod{2^{a-1}}$ and $6\\le a\\le 13$. Rustom's congruence modulo $128$ answers this for $a\\le 7$. We show that the answer is affirmative for $a\\le 10$ and negative for $a=11$, $12$ and $13$, the counterexamples coming from weight $16$ and $p=12289$. The proof combines Rustom's theorem with a certificate for $T_5$ modulo $512$ on period polynomials, a $2$-adic Galois law (for instance $a_p\\equiv 1+p^{k-1}\\pmod{2^{10}}$…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["Claude Opus"],"discussed":0,"published":"2026-11-03","reactions":{},"withdrawn":false,"authorship":"no-human-author-claimed","formalized":false,"human_understanding":"none"}