Singular Q_3 Characters of the Hecke Algebra at Level 163

Correction

The two branches of the local node are not the two nonzero T_2 roots. The full Hecke algebra separates one Q_3 branch off semisimply, and glues the other two.

Using the T_2 factor labels modulo 27, the three singular Q_3 characters are:

Heegner operator values on the three singular characters

operator V1 (mod 3) V5 (mod 3) V7 (mod 3)
2 0 0 0
3 0 0 1
7 2 2 2
11 0 0 0
19 0 0 2
43 1 1 2
67 1 1 2
163 2 2 1

First separation scale: residues modulo 27

operator V1 (mod 27) V5 (mod 27) V7 (mod 27)
2 0 12 3
3 0 18 4
7 2 14 8
11 21 6 12
19 21 12 20
43 7 4 17
67 25 4 14
163 26 26 1

Congruence pattern

The glued local pair is V1_V5.

So the node is formed by the rational branch V1 and one 3-adic branch from the degree-5 orbit. The degree-7 branch is the semisimple Z_3 factor split off by the lifted idempotent from T_3. At the first nontrivial separation scale, T_11 already distinguishes the two branches modulo 27: T_11(V1) = 21 ≡ -6, T_11(V5) = 6, while T_163(V1) = T_163(V5) = 26 ≡ -1. The split branch is characterized by the opposite Atkin–Lehner sign on this local trio: T_163(V7) = 1, so the node is exactly the collision of the two U_163 = -1 branches, with the U_163 = +1 branch split off semisimply.

Honest read

The correct ‘north/south/vacuum’ picture is controlled by the full Hecke algebra, not by T_2 alone. T_2 sees three roots reducing to 0 mod 3, but T_3 separates one of them off semisimply. The remaining two are the actual local branches of the node.