$ \newcommand{\SL}{{\rm SL}} \newcommand{\SU}{{\rm SU}} \newcommand{\GL}{{\rm GL}} \newcommand{\GSp}{{\rm GSp}} \newcommand{\PGSp}{{\rm PGSp}} \newcommand{\SO}{{\rm SO}} \newcommand{\Sp}{{\rm Sp}} \newcommand{\triv}{1} \newcommand{\p}{\mathfrak{p}} \newcommand{\A}{\mathbb{A}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\Z}{\mathbb{Z}} $

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 $\pi\cong\otimes\pi_v$ of $\GSp(4,\A_\Q)$ with the following properties:

• The archimedean component $\pi_\infty$ 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 $\pi_2$ has conductor exponent $\leq4$.
• The non-archimedean components $\pi_p$ for $p\geq3$ are unramified.

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.8.10.0.a		1	IVa	+	$1+2^7T$	$-18360$	0	0	0	1	2	0	2	0	2	1	16	[3,2,1]
G.8.10.0.b		1	X	+	$1+288T+2^{17}T^2$	$-3672$	0	0	0	1	2	0	0	0	0	0	0
G.16.10.0.a		1	XIa	+	$1-2^8T$	$-12888$	0	0	0	0	1	0	0	0	0	0	0
G.16.10.0.b		1	XIa	+	$1+2^8T$	$5928$	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	+	$1-16(-19\pm\sqrt{505})T+2^{17}T^2$	$7248\pm240\sqrt{505}$	0	0	0	0	2	0	0	0	0	0	0
G.16.10.0.f		5	X	+	$1-2^7$$t_{10}$$T+2^{17}T^2$	$\alpha_{10,16}$ (degree 5)	0	0	0	0	5	0	0	0	0	0	0
P.1.10.0.a	1.18.a.a	1	IIb	+	$(1+528T+2^{17}T^2)(1-2^8T)(1-2^9T)$	$21960$	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	+	$(1-2^8T)^2$	$32328$	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	+	$(1-2^8T)(1-2^9T)$	$32040\pm1152\sqrt{114}$	0	0	0	2	2	0	0	0	0	0	0
P.8.10.0.b	4.18.a.a	2	XIa*	+	$1-2^8T$	$23304\pm192\sqrt{9361}$	0	0	0	0	0	0	0	0	0	0	0
P.16.10.0.a	8.18.a.a	2	XIa*	+	$1-2^8T$	$25768\pm256\sqrt{2146}$	0	0	0	0	0	0	0	0	0	0	0
P.16.10.0.b	16.18.a.c	2	XIb	+	$(1-2^8T)(1-2^9T)$	$20448\pm1152\sqrt{114}$	0	0	0	0	2	0	0	0	0	0	0
P.16.10.0.c	16.18.a.d	2	XIb	+	$(1-2^8T)(1-2^9T)$	$26720\pm256\sqrt{2146}$	0	0	0	0	2	0	0	0	0	0	0
$\mathrm{dim}\:S_{10}(\Gamma)$							1	1	2	6	24	4	19	2	9	6	61