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.4.11.0.a | | 1 | X | - | $1+1344T+2^{19}T^2$ | $-13464$ |
0 | 0 | 1 | 2 | 4 |
0 | 1 | 0 | 3 |
0 | 15 | [4,1,1]+[3,3] |
G.8.11.0.a | | 2 | X | - | $1-32(-7\pm\sqrt{55})T+2^{19}T^2$ | $-24(781\pm128\sqrt{55})$ |
0 | 0 | 0 | 2 | 4 |
0 | 0 | 0 | 0 |
0 | 0 | |
G.16.11.0.a | | 1 | X | + | $1+928T+2^{19}T^2$ | $-66096$ |
0 | 0 | 0 | 0 | 1 |
0 | 0 | 0 | 0 |
0 | 0 | |
G.16.11.0.b | | 1 | sc(16) | - | $1$ | $8040$ |
0 | 0 | 0 | 0 | 1 |
0 | 0 | 0 | 1 |
0 | 9 | [2,2,1,1] |
G.16.11.0.c | | 2 | XIa | - | $1-2^9T$ | $24(-1245\pm32\sqrt{21})$ |
0 | 0 | 0 | 0 | 2 |
0 | 0 | 0 | 0 |
0 | 0 | |
G.16.11.0.d | | 2 | XIa | - | $1+2^9T$ | $120(111\pm8\sqrt{69})$ |
0 | 0 | 0 | 0 | 2 |
0 | 0 | 0 | 0 |
0 | 0 | |
G.16.11.0.e | | 1 | IIa or X | - | $1-1152T+2^{19}T^2$ | $-73584$ |
0 | 0 | 0 | 0 | 1 |
0 | 0 | 0 | 0 |
0 | 0 | |
G.16.11.0.f | | 1 | IIa or X | - | $1-256T+2^{19}T^2$ | $18768$ |
0 | 0 | 0 | 0 | 1 |
0 | 0 | 0 | 0 |
0 | 0 | |
G.16.11.0.g | | 1 | IIa or X | - | $1+192T+2^{19}T^2$ | $35568$ |
0 | 0 | 0 | 0 | 1 |
0 | 0 | 0 | 0 |
0 | 0 | |
G.16.11.0.h | | 2 | IIa or X | - | $1-8(-107\pm\sqrt{3961})T+2^{19}T^2$ | $48(425\pm2\sqrt{3961})$ |
0 | 0 | 0 | 0 | 2 |
0 | 0 | 0 | 0 |
0 | 0 | |
G.16.11.0.i | | 4 | IIa or X | - | $1-256\,$$t_{11}$$T+2^{19}T^2$ | $\alpha_{11,16}$ (degree 4) |
0 | 0 | 0 | 0 | 4 |
0 | 0 | 0 | 0 |
0 | 0 | |
P.2.11.0.a | 2.20.a.b | 1 | VIc | - | $(1-2^9T)^2(1-2^{10}T)$ | $25704$ |
0 | 1 | 1 | 2 | 2 |
0 | 0 | 1 | 2 |
1 | 5 | [5,1] |
P.4.11.0.a | 2.20.a.a | 1 | Va* | - | $(1+2^9T)(1-2^9T)(1-2^{10}T)$ | $65640$ |
0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 |
0 | 1 | [1,1,1,1,1,1] |
P.4.11.0.b | 4.20.a.a | 1 | XIb | - | $(1-2^9T)(1-2^{10}T)$ | $78696$ |
0 | 0 | 1 | 1 | 2 |
0 | 0 | 0 | 1 |
0 | 5 | [3,3] |
P.8.11.0.a | 8.20.a.a | 2 | XIb | - | $(1-2^9T)(1-2^{10}T)$ | $24(2699\pm40\sqrt{1453})$ |
0 | 0 | 0 | 2 | 2 |
0 | 0 | 0 | 0 |
0 | 0 | |
P.16.11.0.a | 8.20.a.b | 3 | XIa* | - | $1-2^9T$ | $86643-$$\beta_1$ |
0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 |
0 | 0 | |
P.16.11.0.b | 16.20.a.a | 1 | IIb | - | $(1-2^9T)(1-2^{10}T)$ | $28080$ |
0 | 0 | 0 | 0 | 1 |
0 | 0 | 0 | 0 |
0 | 0 | |
P.16.11.0.c | 16.20.a.b | 1 | XIb | - | $(1-2^9T)(1-2^{10}T)$ | $78768$ |
0 | 0 | 0 | 0 | 1 |
0 | 0 | 0 | 0 |
0 | 0 | |
P.16.11.0.d | 16.20.a.c | 1 | Vb | - | $(1-2^9T)(1-2^{10}T)$ | $91824$ |
0 | 0 | 0 | 0 | 1 |
0 | 0 | 0 | 0 |
0 | 0 | |
P.16.11.0.e | 16.20.a.d | 1 | Vb | - | $(1-2^9T)(1-2^{10}T)$ | $131760$ |
0 | 0 | 0 | 0 | 1 |
0 | 0 | 0 | 0 |
0 | 0 | |
$\mathrm{dim}\:S_{11}(\Gamma)$ |
0 | 1 | 3 | 9 | 33 |
0 | 1 | 1 | 7 |
1 | 35 | |