The following table gives a complete list of the Galois orbits of cuspidal automorphic representations $\pi\cong\otimes\pi_v$ of $\GSp(4,\A_\Q)$ with the following properties:
label | lift from | size | $p=2$ | $p=3$ |
paramodular |
Siegel | Klingen |
Borel | principal |
type | $\varepsilon$ | $L$ | $T(3)$ |
$K(1)$ | $K(2)$ | $K(4)$ | $K(8)$ | $K(16)$ |
$\Gamma_0(2)$ | $\Gamma_0(4)$ | $\Gamma_0'(2)$ | $\Gamma_0'(4)$ |
$B(2)$ | $\Gamma(2)$ | $S_6$ types |
G.4.12.0.a | | 1 | IIIa | + | $1+7\cdot2^8T+2^{20}T^2$ | $-88488$ |
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 | + | $1+2^9T$ | $-14760$ |
0 | 0 | 0 | 1 | 2 |
0 | 2 | 0 | 2 |
1 | 16 | [3,2,1] |
G.8.12.0.b | | 1 | XIa | + | $1-2^{10}T$ | $-229032$ |
0 | 0 | 0 | 1 | 2 |
0 | 1 | 0 | 2 |
0 | 10 | [4,1,1] |
G.8.12.0.c | | 2 | X | + | $1+2^6(12\pm5\sqrt{6})T+2^{21}T^2$ | $504(65\pm64\sqrt{6})$ |
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 | + | $1$ | $72(819\pm64\sqrt{85})$ |
0 | 0 | 0 | 0 | 2 |
0 | 6 | 0 | 1 |
0 | 20 | 2[3,1,1,1] |
G.16.12.0.c | | 2 | XIa | + | $1-2^{10}T$ | $72(-521\pm128\sqrt{5})$ |
0 | 0 | 0 | 0 | 2 |
0 | 0 | 0 | 0 |
0 | 0 | |
G.16.12.0.d | | 2 | XIa | + | $1+2^{10}T$ | $72(831\pm8\sqrt{85})$ |
0 | 0 | 0 | 0 | 2 |
0 | 0 | 0 | 0 |
0 | 0 | |
G.16.12.0.e | | 5 | X | + | $1-2^9$$t_{12,a}$$T+2^{21}T^2$ | $\alpha_{12,16,a}$ |
0 | 0 | 0 | 0 | 5 |
0 | 0 | 0 | 0 |
0 | 0 | |
G.16.12.0.f | | 8 | X | + | $1-2^9$$t_{12,b}$$T+2^{21}T^2$ | $\alpha_{12,16,b}$ |
0 | 0 | 0 | 0 | 8 |
0 | 0 | 0 | 0 |
0 | 0 | |
G.16.12.0.g | | 1 | IIa or X | - | $1+21\cdot2^6T+2^{21}T^2$ | $-185616$ |
0 | 0 | 0 | 0 | 1 |
0 | 0 | 0 | 0 |
0 | 0 | |
P.1.12.0.a | 1.22.a.a | 1 | IIb | + | $(1+288T+2^{21}T^2)(1-2^{10}T)(1-2^{11}T)$ | $107352$ |
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 | + | $(1+2^{10}T)(1-2^{10}T)(1-2^{11}T)$ | $307800$ |
0 | 1 | 1 | 2 | 2 |
1 | 3 | 1 | 2 |
2 | 9 | [4,2] |
P.4.12.0.a | 2.22.a.b | 1 | VIb | + | $(1-2^{10}T)^2$ | $295512$ |
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* | + | $1-2^{10}T$ | $24(11209\pm112\sqrt{2161})$ |
0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 |
0 | 0 | |
P.8.12.0.b | 8.22.a.a | 2 | XIb | + | $(1-2^{10}T)(1-2^{11}T)$ | $24(7645\pm8\sqrt{358549}$ |
0 | 0 | 0 | 2 | 2 |
0 | 0 | 0 | 0 |
0 | 0 | |
P.16.12.0.a | 8.22.a.b | 3 | XIa* | + | $1-2^{10}T$ | $268451-$$\beta_1$ |
0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 |
0 | 0 | |
P.16.12.0.b | 16.22.a.e | 2 | XIb | + | $(1-2^{10}T)(1-2^{11}T)$ | $48(6019\pm4\sqrt{358549}$) |
0 | 0 | 0 | 0 | 2 |
0 | 0 | 0 | 0 |
0 | 0 | |
P.16.12.0.c | 16.22.a.f | 3 | XIb | + | $(1-2^{10}T)(1-2^{11}T)$ | $203941+$$\beta_1$ |
0 | 0 | 0 | 0 | 3 |
0 | 0 | 0 | 0 |
0 | 0 | |
$\mathrm{dim}\:S_{12}(\Gamma)$ |
1 | 2 | 4 | 12 | 45 |
7 | 34 | 4 | 18 |
12 | 120 | |