Embedded newform 1600.4.a.bx.1.1

• Label: 1600.4.a.bx.1.1
• Level: $1600$
• Weight: $4$
• Character: 1600.1
• Self dual: yes
• Analytic conductor: $94.403$
• Analytic rank: $1$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	$N$	$=$	$1600 = 2^{6} \cdot 5^{2}$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	1600.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$94.4030560092$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 10)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1600.1

$q$-expansion

$f(q)$

$=$

$q+8.00000 q^{3} -4.00000 q^{7} +37.0000 q^{9} +12.0000 q^{11} -58.0000 q^{13} -66.0000 q^{17} -100.000 q^{19} -32.0000 q^{21} +132.000 q^{23} +80.0000 q^{27} +90.0000 q^{29} -152.000 q^{31} +96.0000 q^{33} -34.0000 q^{37} -464.000 q^{39} -438.000 q^{41} -32.0000 q^{43} -204.000 q^{47} -327.000 q^{49} -528.000 q^{51} +222.000 q^{53} -800.000 q^{57} +420.000 q^{59} -902.000 q^{61} -148.000 q^{63} +1024.00 q^{67} +1056.00 q^{69} -432.000 q^{71} -362.000 q^{73} -48.0000 q^{77} +160.000 q^{79} -359.000 q^{81} -72.0000 q^{83} +720.000 q^{87} +810.000 q^{89} +232.000 q^{91} -1216.00 q^{93} -1106.00 q^{97} +444.000 q^{99} +O(q^{100})$

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
			$3$	8.00000	1.53960	0.769800	−	0.638285i	$-0.220356\pi$
0.769800	+	0.638285i				$0.220356\pi$
$5$	0	0
			$7$	−4.00000	−0.215980	−0.107990	−	0.994152i	$-0.534441\pi$
−0.107990	+	0.994152i				$0.534441\pi$
$11$	12.0000	0.328921	0.164461	−	0.986384i	$-0.447412\pi$
			0.164461	+	0.986384i	$0.447412\pi$
$13$	−58.0000	−1.23741	−0.618704	−	0.785624i	$-0.712342\pi$
			−0.618704	+	0.785624i	$0.712342\pi$
$17$	−66.0000	−0.941609	−0.470804	−	0.882238i	$-0.656036\pi$
			−0.470804	+	0.882238i	$0.656036\pi$
$19$	−100.000	−1.20745	−0.603726	−	0.797192i	$-0.706318\pi$
			−0.603726	+	0.797192i	$0.706318\pi$
$23$	132.000	1.19669	0.598346	−	0.801238i	$-0.295825\pi$
			0.598346	+	0.801238i	$0.295825\pi$
$29$	90.0000	0.576296	0.288148	−	0.957586i	$-0.406961\pi$
			0.288148	+	0.957586i	$0.406961\pi$
$31$	−152.000	−0.880645	−0.440323	−	0.897840i	$-0.645136\pi$
			−0.440323	+	0.897840i	$0.645136\pi$
$37$	−34.0000	−0.151069	−0.0755347	−	0.997143i	$-0.524066\pi$
			−0.0755347	+	0.997143i	$0.524066\pi$
$41$	−438.000	−1.66839	−0.834196	−	0.551467i	$-0.814068\pi$
			−0.834196	+	0.551467i	$0.814068\pi$
$43$	−32.0000	−0.113487	−0.0567437	−	0.998389i	$-0.518072\pi$
			−0.0567437	+	0.998389i	$0.518072\pi$
$47$	−204.000	−0.633116	−0.316558	−	0.948573i	$-0.602527\pi$
			−0.316558	+	0.948573i	$0.602527\pi$

(See $a_n$ instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1600.4.a.bx.1.1		1
4.3	odd	2		1600.4.a.d.1.1		1
5.4	even	2		320.4.a.b.1.1		1
8.3	odd	2		50.4.a.c.1.1		1
8.5	even	2		400.4.a.b.1.1		1
20.19	odd	2		320.4.a.m.1.1		1
24.11	even	2		450.4.a.q.1.1		1
40.3	even	4		50.4.b.a.49.2		2
40.13	odd	4		400.4.c.c.49.1		2
40.19	odd	2		10.4.a.a.1.1	✓	1
40.27	even	4		50.4.b.a.49.1		2
40.29	even	2		80.4.a.f.1.1		1
40.37	odd	4		400.4.c.c.49.2		2
56.27	even	2		2450.4.a.b.1.1		1
80.19	odd	4		1280.4.d.j.641.2		2
80.29	even	4		1280.4.d.g.641.1		2
80.59	odd	4		1280.4.d.j.641.1		2
80.69	even	4		1280.4.d.g.641.2		2
120.29	odd	2		720.4.a.j.1.1		1
120.59	even	2		90.4.a.a.1.1		1
120.83	odd	4		450.4.c.d.199.1		2
120.107	odd	4		450.4.c.d.199.2		2
280.19	even	6		490.4.e.a.361.1		2
280.59	even	6		490.4.e.a.471.1		2
280.139	even	2		490.4.a.o.1.1		1
280.179	odd	6		490.4.e.i.471.1		2
280.219	odd	6		490.4.e.i.361.1		2
360.59	even	6		810.4.e.w.541.1		2
360.139	odd	6		810.4.e.c.541.1		2
360.259	odd	6		810.4.e.c.271.1		2
360.299	even	6		810.4.e.w.271.1		2
440.219	even	2		1210.4.a.b.1.1		1
520.259	odd	2		1690.4.a.a.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
10.4.a.a.1.1	✓	1	40.19	odd	2
50.4.a.c.1.1		1	8.3	odd	2
50.4.b.a.49.1		2	40.27	even	4
50.4.b.a.49.2		2	40.3	even	4
80.4.a.f.1.1		1	40.29	even	2
90.4.a.a.1.1		1	120.59	even	2
320.4.a.b.1.1		1	5.4	even	2
320.4.a.m.1.1		1	20.19	odd	2
400.4.a.b.1.1		1	8.5	even	2
400.4.c.c.49.1		2	40.13	odd	4
400.4.c.c.49.2		2	40.37	odd	4
450.4.a.q.1.1		1	24.11	even	2
450.4.c.d.199.1		2	120.83	odd	4
450.4.c.d.199.2		2	120.107	odd	4
490.4.a.o.1.1		1	280.139	even	2
490.4.e.a.361.1		2	280.19	even	6
490.4.e.a.471.1		2	280.59	even	6
490.4.e.i.361.1		2	280.219	odd	6
490.4.e.i.471.1		2	280.179	odd	6
720.4.a.j.1.1		1	120.29	odd	2
810.4.e.c.271.1		2	360.259	odd	6
810.4.e.c.541.1		2	360.139	odd	6
810.4.e.w.271.1		2	360.299	even	6
810.4.e.w.541.1		2	360.59	even	6
1210.4.a.b.1.1		1	440.219	even	2
1280.4.d.g.641.1		2	80.29	even	4
1280.4.d.g.641.2		2	80.69	even	4
1280.4.d.j.641.1		2	80.59	odd	4
1280.4.d.j.641.2		2	80.19	odd	4
1600.4.a.d.1.1		1	4.3	odd	2
1600.4.a.bx.1.1		1	1.1	even	1	trivial
1690.4.a.a.1.1		1	520.259	odd	2
2450.4.a.b.1.1		1	56.27	even	2