Automorphic Representations of GSp(4)
Weight 12, conductor dividing 16
The following table gives a complete list of the Galois orbits of cuspidal automorphic representations
π≅⊗πv of
GSp(4,AQ) with the following properties:
- The archimedean component π∞ is a holomorphic discrete series representation with scalar minimal K-type (12,12). These correspond to scalar-valued Siegel modular forms of weight 12.
- The non-archimedean component π2 has conductor exponent ≤4.
- The non-archimedean components πp for p≥3 are unramified.
label | lift from | size | p=2 | p=3 |
paramodular |
Siegel | Klingen |
Borel | principal |
type | ε | L | T(3) |
K(1) | K(2) | K(4) | K(8) | K(16) |
Γ0(2) | Γ0(4) | Γ′0(2) | Γ′0(4) |
B(2) | Γ(2) | S6 types |
G.4.12.0.a | | 1 | IIIa | + | 1+7⋅28T+220T2 | −88488 |
0 | 0 | 1 | 2 | 4 |
2 | 8 | 1 | 5 |
4 | 30 | [4,2]+[3,2,1]+[2,2,2] |
G.8.12.0.a | | 1 | IVa | + | 1+29T | −14760 |
0 | 0 | 0 | 1 | 2 |
0 | 2 | 0 | 2 |
1 | 16 | [3,2,1] |
G.8.12.0.b | | 1 | XIa | + | 1−210T | −229032 |
0 | 0 | 0 | 1 | 2 |
0 | 1 | 0 | 2 |
0 | 10 | [4,1,1] |
G.8.12.0.c | | 2 | X | + | 1+26(12±5√6)T+221T2 | 504(65±64√6) |
0 | 0 | 0 | 2 | 4 |
0 | 0 | 0 | 0 |
0 | 0 | |
G.16.12.0.a | | 1 | VII | + | 1 | −12456 |
0 | 0 | 0 | 0 | 1 |
0 | 4 | 0 | 2 |
0 | 15 | [3,1,1,1]+[2,1,1,1,1] |
G.16.12.0.b | | 2 | IXa | + | 1 | 72(819±64√85) |
0 | 0 | 0 | 0 | 2 |
0 | 6 | 0 | 1 |
0 | 20 | 2[3,1,1,1] |
G.16.12.0.c | | 2 | XIa | + | 1−210T | 72(−521±128√5) |
0 | 0 | 0 | 0 | 2 |
0 | 0 | 0 | 0 |
0 | 0 | |
G.16.12.0.d | | 2 | XIa | + | 1+210T | 72(831±8√85) |
0 | 0 | 0 | 0 | 2 |
0 | 0 | 0 | 0 |
0 | 0 | |
G.16.12.0.e | | 5 | X | + | 1−29t12,aT+221T2 | α12,16,a |
0 | 0 | 0 | 0 | 5 |
0 | 0 | 0 | 0 |
0 | 0 | |
G.16.12.0.f | | 8 | X | + | 1−29t12,bT+221T2 | α12,16,b |
0 | 0 | 0 | 0 | 8 |
0 | 0 | 0 | 0 |
0 | 0 | |
G.16.12.0.g | | 1 | IIa or X | - | 1+21⋅26T+221T2 | −185616 |
0 | 0 | 0 | 0 | 1 |
0 | 0 | 0 | 0 |
0 | 0 | |
P.1.12.0.a | 1.22.a.a | 1 | IIb | + | (1+288T+221T2)(1−210T)(1−211T) | 107352 |
1 | 1 | 2 | 2 | 3 |
3 | 7 | 2 | 4 |
4 | 15 | [6]+[4,2]+[2,2,2] |
P.2.12.0.a | 2.22.a.a | 1 | Vb | + | (1+210T)(1−210T)(1−211T) | 307800 |
0 | 1 | 1 | 2 | 2 |
1 | 3 | 1 | 2 |
2 | 9 | [4,2] |
P.4.12.0.a | 2.22.a.b | 1 | VIb | + | (1−210T)2 | 295512 |
0 | 0 | 0 | 0 | 0 |
1 | 3 | 0 | 0 |
1 | 5 | [2,2,2] |
P.8.12.0.a | 4.22.a.a | 2 | XIa* | + | 1−210T | 24(11209±112√2161) |
0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 |
0 | 0 | |
P.8.12.0.b | 8.22.a.a | 2 | XIb | + | (1−210T)(1−211T) | 24(7645±8√358549 |
0 | 0 | 0 | 2 | 2 |
0 | 0 | 0 | 0 |
0 | 0 | |
P.16.12.0.a | 8.22.a.b | 3 | XIa* | + | 1−210T | 268451−β1 |
0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 |
0 | 0 | |
P.16.12.0.b | 16.22.a.e | 2 | XIb | + | (1−210T)(1−211T) | 48(6019±4√358549) |
0 | 0 | 0 | 0 | 2 |
0 | 0 | 0 | 0 |
0 | 0 | |
P.16.12.0.c | 16.22.a.f | 3 | XIb | + | (1−210T)(1−211T) | 203941+β1 |
0 | 0 | 0 | 0 | 3 |
0 | 0 | 0 | 0 |
0 | 0 | |
dimS12(Γ) |
1 | 2 | 4 | 12 | 45 |
7 | 34 | 4 | 18 |
12 | 120 | |