Automorphic Representations of GSp(4)
Weight 10, 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 (10,10). These correspond to scalar-valued Siegel modular forms of weight 10.
- 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.8.10.0.a | | 1 | IVa | + | 1+27T | −18360 |
0 | 0 | 0 | 1 | 2 |
0 | 2 | 0 | 2 |
1 | 16 | [3,2,1] |
G.8.10.0.b | | 1 | X | + | 1+288T+217T2 | −3672 |
0 | 0 | 0 | 1 | 2 |
0 | 0 | 0 | 0 |
0 | 0 | |
G.16.10.0.a | | 1 | XIa | + | 1−28T | −12888 |
0 | 0 | 0 | 0 | 1 |
0 | 0 | 0 | 0 |
0 | 0 | |
G.16.10.0.b | | 1 | XIa | + | 1+28T | 5928 |
0 | 0 | 0 | 0 | 1 |
0 | 0 | 0 | 0 |
0 | 0 | |
G.16.10.0.c | | 1 | IXa | + | 1 | −3768 |
0 | 0 | 0 | 0 | 1 |
0 | 3 | 0 | 1 |
0 | 10 | [3,1,1,1] |
G.16.10.0.d | | 1 | VII | + | 1 | −1080 |
0 | 0 | 0 | 0 | 1 |
0 | 4 | 0 | 2 |
0 | 15 | [3,1,1,1]+[2,1,1,1,1] |
G.16.10.0.e | | 2 | X | + | 1−16(−19±√505)T+217T2 | 7248±240√505 |
0 | 0 | 0 | 0 | 2 |
0 | 0 | 0 | 0 |
0 | 0 | |
G.16.10.0.f | | 5 | X | + | 1−27t10T+217T2 | α10,16 (degree 5) |
0 | 0 | 0 | 0 | 5 |
0 | 0 | 0 | 0 |
0 | 0 | |
P.1.10.0.a | 1.18.a.a | 1 | IIb | + | (1+528T+217T2)(1−28T)(1−29T) | 21960 |
1 | 1 | 2 | 2 | 3 |
3 | 7 | 2 | 4 |
4 | 15 | [6]+[4,2]+[2,2,2] |
P.4.10.0.a | 2.18.a.a | 1 | VIb | + | (1−28T)2 | 32328 |
0 | 0 | 0 | 0 | 0 |
1 | 3 | 0 | 0 |
1 | 5 | [2,2,2] |
P.8.10.0.a | 8.18.a.b | 2 | XIb | + | (1−28T)(1−29T) | 32040±1152√114 |
0 | 0 | 0 | 2 | 2 |
0 | 0 | 0 | 0 |
0 | 0 | |
P.8.10.0.b | 4.18.a.a | 2 | XIa* | + | 1−28T | 23304±192√9361 |
0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 |
0 | 0 | |
P.16.10.0.a | 8.18.a.a | 2 | XIa* | + | 1−28T | 25768±256√2146 |
0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 |
0 | 0 | |
P.16.10.0.b | 16.18.a.c | 2 | XIb | + | (1−28T)(1−29T) | 20448±1152√114 |
0 | 0 | 0 | 0 | 2 |
0 | 0 | 0 | 0 |
0 | 0 | |
P.16.10.0.c | 16.18.a.d | 2 | XIb | + | (1−28T)(1−29T) | 26720±256√2146 |
0 | 0 | 0 | 0 | 2 |
0 | 0 | 0 | 0 |
0 | 0 | |
dimS10(Γ) |
1 | 1 | 2 | 6 | 24 |
4 | 19 | 2 | 9 |
6 | 61 | |