Embedded newform 1089.4.a.g.1.1

• Label: 1089.4.a.g.1.1
• Level: $1089$
• Weight: $4$
• Character: 1089.1
• Self dual: yes
• Analytic conductor: $64.253$
• Analytic rank: $1$
• Dimension: $1$
• CM discriminant: -3
• Inner twists: $2$

Newspace parameters

Level:	$N$	$=$	$1089 = 3^{2} \cdot 11^{2}$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	1089.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$64.2530799963$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 9)
Fricke sign:	$-1$
Sato-Tate group:	$N(\mathrm{U}(1))$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1089.1

$q$-expansion

$f(q)$

$=$

$q-8.00000 q^{4} -20.0000 q^{7} +70.0000 q^{13} +64.0000 q^{16} -56.0000 q^{19} -125.000 q^{25} +160.000 q^{28} +308.000 q^{31} +110.000 q^{37} +520.000 q^{43} +57.0000 q^{49} -560.000 q^{52} -182.000 q^{61} -512.000 q^{64} -880.000 q^{67} -1190.00 q^{73} +448.000 q^{76} -884.000 q^{79} -1400.00 q^{91} -1330.00 q^{97} +O(q^{100})$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

(See $a_n$ instead)

$p$	$a_p$	$a_p / p^{(k-1)/2}$	$\alpha_p$			$\theta_p$
$2$	0	0		−	1.00000i	$-0.5\pi$
					1.00000i	$0.5\pi$
$3$	0	0
			$5$	0	0		−	1.00000i	$-0.5\pi$
		1.00000i				$0.5\pi$
$7$	−20.0000	−1.07990	−0.539949	−	0.841698i	$-0.681557\pi$
			−0.539949	+	0.841698i	$0.681557\pi$
$11$	0	0
			$13$	70.0000	1.49342	0.746712	−	0.665148i	$-0.231631\pi$
0.746712	+	0.665148i				$0.231631\pi$
$17$	0	0		−	1.00000i	$-0.5\pi$
					1.00000i	$0.5\pi$
$19$	−56.0000	−0.676173	−0.338086	−	0.941115i	$-0.609780\pi$
			−0.338086	+	0.941115i	$0.609780\pi$
$23$	0	0		−	1.00000i	$-0.5\pi$
					1.00000i	$0.5\pi$
$29$	0	0		−	1.00000i	$-0.5\pi$
					1.00000i	$0.5\pi$
$31$	308.000	1.78447	0.892233	−	0.451576i	$-0.149138\pi$
			0.892233	+	0.451576i	$0.149138\pi$
$37$	110.000	0.488754	0.244377	−	0.969680i	$-0.421417\pi$
			0.244377	+	0.969680i	$0.421417\pi$
$41$	0	0		−	1.00000i	$-0.5\pi$
					1.00000i	$0.5\pi$
$43$	520.000	1.84417	0.922084	−	0.386989i	$-0.126485\pi$
			0.922084	+	0.386989i	$0.126485\pi$
$47$	0	0		−	1.00000i	$-0.5\pi$
					1.00000i	$0.5\pi$

(See $a_n$ instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1089.4.a.g.1.1		1
3.2	odd	2	CM	1089.4.a.g.1.1		1
11.10	odd	2		9.4.a.a.1.1	✓	1
33.32	even	2		9.4.a.a.1.1	✓	1
44.43	even	2		144.4.a.d.1.1		1
55.32	even	4		225.4.b.g.199.2		2
55.43	even	4		225.4.b.g.199.1		2
55.54	odd	2		225.4.a.d.1.1		1
77.10	even	6		441.4.e.j.226.1		2
77.32	odd	6		441.4.e.i.226.1		2
77.54	even	6		441.4.e.j.361.1		2
77.65	odd	6		441.4.e.i.361.1		2
77.76	even	2		441.4.a.f.1.1		1
88.21	odd	2		576.4.a.m.1.1		1
88.43	even	2		576.4.a.l.1.1		1
99.32	even	6		81.4.c.b.55.1		2
99.43	odd	6		81.4.c.b.28.1		2
99.65	even	6		81.4.c.b.28.1		2
99.76	odd	6		81.4.c.b.55.1		2
132.131	odd	2		144.4.a.d.1.1		1
143.142	odd	2		1521.4.a.g.1.1		1
165.32	odd	4		225.4.b.g.199.2		2
165.98	odd	4		225.4.b.g.199.1		2
165.164	even	2		225.4.a.d.1.1		1
231.32	even	6		441.4.e.i.226.1		2
231.65	even	6		441.4.e.i.361.1		2
231.131	odd	6		441.4.e.j.361.1		2
231.164	odd	6		441.4.e.j.226.1		2
231.230	odd	2		441.4.a.f.1.1		1
264.131	odd	2		576.4.a.l.1.1		1
264.197	even	2		576.4.a.m.1.1		1
429.428	even	2		1521.4.a.g.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9.4.a.a.1.1	✓	1	11.10	odd	2
9.4.a.a.1.1	✓	1	33.32	even	2
81.4.c.b.28.1		2	99.43	odd	6
81.4.c.b.28.1		2	99.65	even	6
81.4.c.b.55.1		2	99.32	even	6
81.4.c.b.55.1		2	99.76	odd	6
144.4.a.d.1.1		1	44.43	even	2
144.4.a.d.1.1		1	132.131	odd	2
225.4.a.d.1.1		1	55.54	odd	2
225.4.a.d.1.1		1	165.164	even	2
225.4.b.g.199.1		2	55.43	even	4
225.4.b.g.199.1		2	165.98	odd	4
225.4.b.g.199.2		2	55.32	even	4
225.4.b.g.199.2		2	165.32	odd	4
441.4.a.f.1.1		1	77.76	even	2
441.4.a.f.1.1		1	231.230	odd	2
441.4.e.i.226.1		2	77.32	odd	6
441.4.e.i.226.1		2	231.32	even	6
441.4.e.i.361.1		2	77.65	odd	6
441.4.e.i.361.1		2	231.65	even	6
441.4.e.j.226.1		2	77.10	even	6
441.4.e.j.226.1		2	231.164	odd	6
441.4.e.j.361.1		2	77.54	even	6
441.4.e.j.361.1		2	231.131	odd	6
576.4.a.l.1.1		1	88.43	even	2
576.4.a.l.1.1		1	264.131	odd	2
576.4.a.m.1.1		1	88.21	odd	2
576.4.a.m.1.1		1	264.197	even	2
1089.4.a.g.1.1		1	1.1	even	1	trivial
1089.4.a.g.1.1		1	3.2	odd	2	CM
1521.4.a.g.1.1		1	143.142	odd	2
1521.4.a.g.1.1		1	429.428	even	2