Embedded newform 8820.2.a.w.1.1

• Label: 8820.2.a.w.1.1
• Level: $8820$
• Weight: $2$
• Character: 8820.1
• Self dual: yes
• Analytic conductor: $70.428$
• Analytic rank: $0$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	$N$	$=$	$8820 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	8820.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$70.4280545828$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 140)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	8820.1

$q$-expansion

$f(q)$

$=$

$q+1.00000 q^{5} +2.00000 q^{11} -6.00000 q^{13} -2.00000 q^{17} +9.00000 q^{23} +1.00000 q^{25} -3.00000 q^{29} +2.00000 q^{31} +8.00000 q^{37} -5.00000 q^{41} +1.00000 q^{43} -8.00000 q^{47} -4.00000 q^{53} +2.00000 q^{55} +8.00000 q^{59} +7.00000 q^{61} -6.00000 q^{65} -3.00000 q^{67} -8.00000 q^{71} +14.0000 q^{73} +4.00000 q^{79} +1.00000 q^{83} -2.00000 q^{85} -13.0000 q^{89} -10.0000 q^{97} +O(q^{100})$

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
			$3$	0	0
$5$	1.00000	0.447214
			$7$	0	0
$11$	2.00000	0.603023				0.301511	−	0.953463i	$-0.402509\pi$
			0.301511	+	0.953463i	$0.402509\pi$
$13$	−6.00000	−1.66410	−0.832050	−	0.554700i	$-0.812833\pi$
			−0.832050	+	0.554700i	$0.812833\pi$
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
			−0.242536	+	0.970143i	$0.577979\pi$
$19$	0	0		−	1.00000i	$-0.5\pi$
					1.00000i	$0.5\pi$
$23$	9.00000	1.87663	0.938315	−	0.345782i	$-0.112386\pi$
			0.938315	+	0.345782i	$0.112386\pi$
$29$	−3.00000	−0.557086	−0.278543	−	0.960424i	$-0.589851\pi$
			−0.278543	+	0.960424i	$0.589851\pi$
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
			0.179605	+	0.983739i	$0.442518\pi$
$37$	8.00000	1.31519	0.657596	−	0.753371i	$-0.271573\pi$
			0.657596	+	0.753371i	$0.271573\pi$
$41$	−5.00000	−0.780869	−0.390434	−	0.920631i	$-0.627675\pi$
			−0.390434	+	0.920631i	$0.627675\pi$
$43$	1.00000	0.152499	0.0762493	−	0.997089i	$-0.475706\pi$
			0.0762493	+	0.997089i	$0.475706\pi$
$47$	−8.00000	−1.16692	−0.583460	−	0.812142i	$-0.698301\pi$
			−0.583460	+	0.812142i	$0.698301\pi$

(See $a_n$ instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8820.2.a.w.1.1		1
3.2	odd	2		980.2.a.a.1.1		1
7.2	even	3		1260.2.s.b.361.1		2
7.4	even	3		1260.2.s.b.541.1		2
7.6	odd	2		8820.2.a.k.1.1		1
12.11	even	2		3920.2.a.bi.1.1		1
15.2	even	4		4900.2.e.c.2549.2		2
15.8	even	4		4900.2.e.c.2549.1		2
15.14	odd	2		4900.2.a.v.1.1		1
21.2	odd	6		140.2.i.b.81.1	✓	2
21.5	even	6		980.2.i.a.361.1		2
21.11	odd	6		140.2.i.b.121.1	yes	2
21.17	even	6		980.2.i.a.961.1		2
21.20	even	2		980.2.a.i.1.1		1
84.11	even	6		560.2.q.a.401.1		2
84.23	even	6		560.2.q.a.81.1		2
84.83	odd	2		3920.2.a.d.1.1		1
105.2	even	12		700.2.r.c.249.1		4
105.23	even	12		700.2.r.c.249.2		4
105.32	even	12		700.2.r.c.149.2		4
105.44	odd	6		700.2.i.a.501.1		2
105.53	even	12		700.2.r.c.149.1		4
105.62	odd	4		4900.2.e.b.2549.1		2
105.74	odd	6		700.2.i.a.401.1		2
105.83	odd	4		4900.2.e.b.2549.2		2
105.104	even	2		4900.2.a.a.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
140.2.i.b.81.1	✓	2	21.2	odd	6
140.2.i.b.121.1	yes	2	21.11	odd	6
560.2.q.a.81.1		2	84.23	even	6
560.2.q.a.401.1		2	84.11	even	6
700.2.i.a.401.1		2	105.74	odd	6
700.2.i.a.501.1		2	105.44	odd	6
700.2.r.c.149.1		4	105.53	even	12
700.2.r.c.149.2		4	105.32	even	12
700.2.r.c.249.1		4	105.2	even	12
700.2.r.c.249.2		4	105.23	even	12
980.2.a.a.1.1		1	3.2	odd	2
980.2.a.i.1.1		1	21.20	even	2
980.2.i.a.361.1		2	21.5	even	6
980.2.i.a.961.1		2	21.17	even	6
1260.2.s.b.361.1		2	7.2	even	3
1260.2.s.b.541.1		2	7.4	even	3
3920.2.a.d.1.1		1	84.83	odd	2
3920.2.a.bi.1.1		1	12.11	even	2
4900.2.a.a.1.1		1	105.104	even	2
4900.2.a.v.1.1		1	15.14	odd	2
4900.2.e.b.2549.1		2	105.62	odd	4
4900.2.e.b.2549.2		2	105.83	odd	4
4900.2.e.c.2549.1		2	15.8	even	4
4900.2.e.c.2549.2		2	15.2	even	4
8820.2.a.k.1.1		1	7.6	odd	2
8820.2.a.w.1.1		1	1.1	even	1	trivial