Embedded newform 800.5.e.e.399.20

• Label: 800.5.e.e.399.20
• Level: $800$
• Weight: $5$
• Character: 800.399
• Analytic conductor: $82.696$
• Analytic rank: $0$
• Dimension: $32$
• Inner twists: $4$

Newspace parameters

Level:	$N$	$=$	$800 = 2^{5} \cdot 5^{2}$
Weight:	$k$	$=$	$5$
Character orbit:	$[\chi]$	$=$	800.e (of order $2$, degree $1$, not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$82.6959704671$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	no (minimal twist has level 40)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			399.20
Character	$\chi$	$=$	800.399
Dual form			800.5.e.e.399.19

$q$-expansion

$f(q)$	$=$	$q+0.715641i q^{3} -25.0983 q^{7} +80.4879 q^{9} +156.704 q^{11} -301.788 q^{13} +123.427i q^{17} +322.113 q^{19} -17.9613i q^{21} -45.9062 q^{23} +115.567i q^{27} -1084.15i q^{29} +1624.75i q^{31} +112.144i q^{33} +895.053 q^{37} -215.972i q^{39} -1097.66 q^{41} -950.337i q^{43} +1343.83 q^{47} -1771.08 q^{49} -88.3294 q^{51} -709.918 q^{53} +230.517i q^{57} +4466.60 q^{59} -933.943i q^{61} -2020.10 q^{63} -4019.28i q^{67} -32.8523i q^{69} -3551.94i q^{71} +5298.05i q^{73} -3933.01 q^{77} +11461.9i q^{79} +6436.81 q^{81} +6588.95i q^{83} +775.861 q^{87} +6280.53 q^{89} +7574.35 q^{91} -1162.73 q^{93} +14960.7i q^{97} +12612.8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$32 q - 864 q^{9} - 384 q^{11} - 1408 q^{19} - 4416 q^{41} + 4960 q^{49} + 35584 q^{51} + 28032 q^{59} + 20768 q^{81} - 13632 q^{89} - 49152 q^{91} + 5248 q^{99}+O(q^{100})$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/800\mathbb{Z}\right)^\times$.

$n$	$101$	$351$	$577$
$\chi(n)$	$-1$	$-1$	$-1$

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
			$3$	0.715641i	0.0795156i	0.999209	+	0.0397578i	$0.0126586\pi$
−0.999209	+	0.0397578i				$0.987341\pi$
$5$	0	0
			$7$	−25.0983	−0.512209	−0.256105	−	0.966649i	$-0.582439\pi$
−0.256105	+	0.966649i				$0.582439\pi$
$11$	156.704	1.29508	0.647539	−	0.762033i	$-0.275798\pi$
			0.647539	+	0.762033i	$0.275798\pi$
$13$	−301.788	−1.78573	−0.892863	−	0.450328i	$-0.851307\pi$
			−0.892863	+	0.450328i	$0.851307\pi$
$17$	123.427i	0.427083i	0.976934	+	0.213542i	$0.0684999\pi$
			−0.976934	+	0.213542i	$0.931500\pi$
$19$	322.113	0.892280	0.446140	−	0.894963i	$-0.352798\pi$
			0.446140	+	0.894963i	$0.352798\pi$
$23$	−45.9062	−0.0867792	−0.0433896	−	0.999058i	$-0.513816\pi$
			−0.0433896	+	0.999058i	$0.513816\pi$
$29$	− 1084.15i	− 1.28912i	−0.764554	−	0.644560i	$-0.777041\pi$
			0.764554	−	0.644560i	$-0.222959\pi$
$31$	1624.75i	1.69068i	0.534227	+	0.845341i	$0.320603\pi$
			−0.534227	+	0.845341i	$0.679397\pi$
$37$	895.053	0.653801	0.326900	−	0.945059i	$-0.393996\pi$
			0.326900	+	0.945059i	$0.393996\pi$
$41$	−1097.66	−0.652978	−0.326489	−	0.945201i	$-0.605866\pi$
			−0.326489	+	0.945201i	$0.605866\pi$
$43$	− 950.337i	− 0.513974i	−0.966415	−	0.256987i	$-0.917270\pi$
			0.966415	−	0.256987i	$-0.0827297\pi$
$47$	1343.83	0.608341	0.304171	−	0.952618i	$-0.401621\pi$
			0.304171	+	0.952618i	$0.401621\pi$

(See $a_n$ instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	800.5.e.e.399.20		32
4.3	odd	2		200.5.e.e.99.18		32
5.2	odd	4		160.5.g.a.111.8		16
5.3	odd	4		800.5.g.h.751.10		16
5.4	even	2	inner	800.5.e.e.399.13		32
8.3	odd	2	inner	800.5.e.e.399.14		32
8.5	even	2		200.5.e.e.99.16		32
15.2	even	4		1440.5.g.a.271.4		16
20.3	even	4		200.5.g.h.51.15		16
20.7	even	4		40.5.g.a.11.2	yes	16
20.19	odd	2		200.5.e.e.99.15		32
40.3	even	4		800.5.g.h.751.9		16
40.13	odd	4		200.5.g.h.51.16		16
40.19	odd	2	inner	800.5.e.e.399.19		32
40.27	even	4		160.5.g.a.111.7		16
40.29	even	2		200.5.e.e.99.17		32
40.37	odd	4		40.5.g.a.11.1	✓	16
60.47	odd	4		360.5.g.a.91.15		16
120.77	even	4		360.5.g.a.91.16		16
120.107	odd	4		1440.5.g.a.271.13		16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.5.g.a.11.1	✓	16	40.37	odd	4
40.5.g.a.11.2	yes	16	20.7	even	4
160.5.g.a.111.7		16	40.27	even	4
160.5.g.a.111.8		16	5.2	odd	4
200.5.e.e.99.15		32	20.19	odd	2
200.5.e.e.99.16		32	8.5	even	2
200.5.e.e.99.17		32	40.29	even	2
200.5.e.e.99.18		32	4.3	odd	2
200.5.g.h.51.15		16	20.3	even	4
200.5.g.h.51.16		16	40.13	odd	4
360.5.g.a.91.15		16	60.47	odd	4
360.5.g.a.91.16		16	120.77	even	4
800.5.e.e.399.13		32	5.4	even	2	inner
800.5.e.e.399.14		32	8.3	odd	2	inner
800.5.e.e.399.19		32	40.19	odd	2	inner
800.5.e.e.399.20		32	1.1	even	1	trivial
800.5.g.h.751.9		16	40.3	even	4
800.5.g.h.751.10		16	5.3	odd	4
1440.5.g.a.271.4		16	15.2	even	4
1440.5.g.a.271.13		16	120.107	odd	4