D₄ · Neutral gradings
12 neutral gradings · KL, Fourier, Arthur packets
| Ambient orbit | KL block sizes | Arthur parameters | Parameters in packets | Graded orbits | Local systems |
|---|---|---|---|---|---|
| 11111111 | 1 × 1 | 1 | 1 | 1 | 1 |
| 221111 | 2 × 2 | 2 | 2 | 2 | 2 |
| 2222+ | 3 × 3, 1 × 1 | 3 | 4 | 3 | 4 |
| 2222- | 3 × 3, 1 × 1 | 3 | 4 | 3 | 4 |
| 311111 | 3 × 3, 1 × 1 | 3 | 4 | 3 | 4 |
| 3221 | 8 × 8, 2 × 2, 2 × 2, 2 × 2 | 8 | 14 | 8 | 14 |
| 3311 | 8 × 8, 2 × 2, 2 × 2, 2 × 2 | 7 | 14 | 7 | 14 |
| 44+ | 6 × 6, 2 × 2 | 2 | 4 | 6 | 8 |
| 44- | 6 × 6, 2 × 2 | 2 | 4 | 6 | 8 |
| 5111 | 6 × 6, 2 × 2 | 2 | 4 | 6 | 8 |
| 53 | 15 × 15, 4 × 4, 4 × 4, 4 × 4 | 8 | 20 | 15 | 27 |
| 71 | 16 × 16, 4 × 4, 4 × 4, 4 × 4 | 2 | 8 | 16 | 28 |
Counting conventions
Matrix dimensions are listed separately for each block. Parameters in packets counts distinct IC parameters, without multiplicities or repeated membership. Local systems counts orbit–local-system pairs across all exported series. “Not computed” means that the selected export does not contain that result, rather than a count of zero.