Embedded newform 200.2.a.b.1.1

• Label: 200.2.a.b.1.1
• Level: $200$
• Weight: $2$
• Character: 200.1
• Self dual: yes
• Analytic conductor: $1.597$
• Analytic rank: $1$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	$N$	$=$	$200 = 2^{3} \cdot 5^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	200.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$1.59700804043$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 40)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	200.1

$q$-expansion

$f(q)$

$=$

$q-2.00000 q^{3} -2.00000 q^{7} +1.00000 q^{9} -4.00000 q^{11} -4.00000 q^{13} -4.00000 q^{19} +4.00000 q^{21} +2.00000 q^{23} +4.00000 q^{27} +2.00000 q^{29} +8.00000 q^{33} -4.00000 q^{37} +8.00000 q^{39} +2.00000 q^{41} +6.00000 q^{43} +6.00000 q^{47} -3.00000 q^{49} +4.00000 q^{53} +8.00000 q^{57} -12.0000 q^{59} -10.0000 q^{61} -2.00000 q^{63} -14.0000 q^{67} -4.00000 q^{69} +8.00000 q^{71} -8.00000 q^{73} +8.00000 q^{77} +16.0000 q^{79} -11.0000 q^{81} -2.00000 q^{83} -4.00000 q^{87} +6.00000 q^{89} +8.00000 q^{91} -16.0000 q^{97} -4.00000 q^{99} +O(q^{100})$

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
			$3$	−2.00000	−1.15470	−0.577350	−	0.816497i	$-0.695913\pi$
−0.577350	+	0.816497i				$0.695913\pi$
$5$	0	0
			$7$	−2.00000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
−0.377964	+	0.925820i				$0.623376\pi$
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
			−0.603023	+	0.797724i	$0.706037\pi$
$13$	−4.00000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
			−0.554700	+	0.832050i	$0.687167\pi$
$17$	0	0		−	1.00000i	$-0.5\pi$
					1.00000i	$0.5\pi$
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
			−0.458831	+	0.888523i	$0.651732\pi$
$23$	2.00000	0.417029	0.208514	−	0.978019i	$-0.433137\pi$
			0.208514	+	0.978019i	$0.433137\pi$
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
			0.185695	+	0.982607i	$0.440546\pi$
$31$	0	0		−	1.00000i	$-0.5\pi$
					1.00000i	$0.5\pi$
$37$	−4.00000	−0.657596	−0.328798	−	0.944400i	$-0.606644\pi$
			−0.328798	+	0.944400i	$0.606644\pi$
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
			0.156174	+	0.987730i	$0.450084\pi$
$43$	6.00000	0.914991	0.457496	−	0.889212i	$-0.348747\pi$
			0.457496	+	0.889212i	$0.348747\pi$
$47$	6.00000	0.875190	0.437595	−	0.899172i	$-0.355830\pi$
			0.437595	+	0.899172i	$0.355830\pi$

(See $a_n$ instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	200.2.a.b.1.1		1
3.2	odd	2		1800.2.a.j.1.1		1
4.3	odd	2		400.2.a.g.1.1		1
5.2	odd	4		40.2.c.a.9.2	yes	2
5.3	odd	4		40.2.c.a.9.1	✓	2
5.4	even	2		200.2.a.d.1.1		1
7.6	odd	2		9800.2.a.bf.1.1		1
8.3	odd	2		1600.2.a.d.1.1		1
8.5	even	2		1600.2.a.v.1.1		1
12.11	even	2		3600.2.a.bb.1.1		1
15.2	even	4		360.2.f.c.289.2		2
15.8	even	4		360.2.f.c.289.1		2
15.14	odd	2		1800.2.a.s.1.1		1
20.3	even	4		80.2.c.a.49.2		2
20.7	even	4		80.2.c.a.49.1		2
20.19	odd	2		400.2.a.b.1.1		1
35.13	even	4		1960.2.g.b.1569.2		2
35.27	even	4		1960.2.g.b.1569.1		2
35.34	odd	2		9800.2.a.d.1.1		1
40.3	even	4		320.2.c.b.129.1		2
40.13	odd	4		320.2.c.c.129.2		2
40.19	odd	2		1600.2.a.u.1.1		1
40.27	even	4		320.2.c.b.129.2		2
40.29	even	2		1600.2.a.f.1.1		1
40.37	odd	4		320.2.c.c.129.1		2
60.23	odd	4		720.2.f.e.289.1		2
60.47	odd	4		720.2.f.e.289.2		2
60.59	even	2		3600.2.a.k.1.1		1
80.3	even	4		1280.2.f.b.129.2		2
80.13	odd	4		1280.2.f.f.129.2		2
80.27	even	4		1280.2.f.b.129.1		2
80.37	odd	4		1280.2.f.f.129.1		2
80.43	even	4		1280.2.f.e.129.1		2
80.53	odd	4		1280.2.f.a.129.1		2
80.67	even	4		1280.2.f.e.129.2		2
80.77	odd	4		1280.2.f.a.129.2		2
120.53	even	4		2880.2.f.h.1729.2		2
120.77	even	4		2880.2.f.h.1729.1		2
120.83	odd	4		2880.2.f.i.1729.2		2
120.107	odd	4		2880.2.f.i.1729.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.2.c.a.9.1	✓	2	5.3	odd	4
40.2.c.a.9.2	yes	2	5.2	odd	4
80.2.c.a.49.1		2	20.7	even	4
80.2.c.a.49.2		2	20.3	even	4
200.2.a.b.1.1		1	1.1	even	1	trivial
200.2.a.d.1.1		1	5.4	even	2
320.2.c.b.129.1		2	40.3	even	4
320.2.c.b.129.2		2	40.27	even	4
320.2.c.c.129.1		2	40.37	odd	4
320.2.c.c.129.2		2	40.13	odd	4
360.2.f.c.289.1		2	15.8	even	4
360.2.f.c.289.2		2	15.2	even	4
400.2.a.b.1.1		1	20.19	odd	2
400.2.a.g.1.1		1	4.3	odd	2
720.2.f.e.289.1		2	60.23	odd	4
720.2.f.e.289.2		2	60.47	odd	4
1280.2.f.a.129.1		2	80.53	odd	4
1280.2.f.a.129.2		2	80.77	odd	4
1280.2.f.b.129.1		2	80.27	even	4
1280.2.f.b.129.2		2	80.3	even	4
1280.2.f.e.129.1		2	80.43	even	4
1280.2.f.e.129.2		2	80.67	even	4
1280.2.f.f.129.1		2	80.37	odd	4
1280.2.f.f.129.2		2	80.13	odd	4
1600.2.a.d.1.1		1	8.3	odd	2
1600.2.a.f.1.1		1	40.29	even	2
1600.2.a.u.1.1		1	40.19	odd	2
1600.2.a.v.1.1		1	8.5	even	2
1800.2.a.j.1.1		1	3.2	odd	2
1800.2.a.s.1.1		1	15.14	odd	2
1960.2.g.b.1569.1		2	35.27	even	4
1960.2.g.b.1569.2		2	35.13	even	4
2880.2.f.h.1729.1		2	120.77	even	4
2880.2.f.h.1729.2		2	120.53	even	4
2880.2.f.i.1729.1		2	120.107	odd	4
2880.2.f.i.1729.2		2	120.83	odd	4
3600.2.a.k.1.1		1	60.59	even	2
3600.2.a.bb.1.1		1	12.11	even	2
9800.2.a.d.1.1		1	35.34	odd	2
9800.2.a.bf.1.1		1	7.6	odd	2