This scan records the currently completed 3-adic local T_2 signatures in the comparison family.
| level | status | root counts mod 3,9,27,81 |
linear Q3 factors | degree multiset over Q_3 |
|---|---|---|---|---|
163 |
canonical |
[1, 3, 9, 18] |
3 |
[1, 1, 1, 4, 6] |
829 |
canonical |
[2, 6, 9, 18] |
3 |
[1, 1, 1, 2, 3, 4, 9, 14, 33] |
2053 |
canonical |
[1, 3, 9, 9] |
1 |
[1, 2, 2, 2, 4, 5, 6, 8, 28, 32, 80] |
Pending full Q_3 factorization: [3853, 4933].
163: linear root residues [{'3': 0, '9': 3, '27': 12, '81': 12}, {'3': 0, '9': 3, '27': 3, '81': 30}, {'3': 0, '9': 0, '27': 0, '81': 0}]829: linear root residues [{'3': 0, '9': 3, '27': 12, '81': 12}, {'3': 0, '9': 0, '27': 0, '81': 27}, {'3': 0, '9': 0, '27': 0, '81': 0}]2053: linear root residues [{'3': 0, '9': 3, '27': 21, '81': 75}]163: basis determinant valuation 1, T_3 mod 27 = [0, 1, 0], T_11 mod 27 = [0, 0, 1], T_163 mod 27 = [8, 14, 24].829: not available, because T_2 is not primitive enough over Q to recover Hecke operators from its powers.2053: not available, because level does not have exactly three linear Q_3 factors.The enlarged canonical mod-3 class is not uniform at the Q_3 level. In particular, if one canonical level already has fewer than three linear Q_3 factors, then the exact 163-style three-branch lift is a stricter subclass than mod-3 canonicity.
The nearest false positives [3853, 4933] have not finished their full Q_3 factorization yet, so this note currently certifies only the split inside the canonical sample.