Embedded newform 800.5.e.e.399.10

• Label: 800.5.e.e.399.10
• 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			399.10
Character	$\chi$	$=$	800.399
Dual form			800.5.e.e.399.9

$q$-expansion

$f(q)$	$=$	$q+10.2034i q^{3} +43.3025 q^{7} -23.1094 q^{9} +165.050 q^{11} -201.647 q^{13} -172.778i q^{17} -640.745 q^{19} +441.833i q^{21} +497.626 q^{23} +590.681i q^{27} +509.953i q^{29} +492.136i q^{31} +1684.07i q^{33} +678.519 q^{37} -2057.49i q^{39} -613.848 q^{41} +1392.38i q^{43} +965.658 q^{47} -525.894 q^{49} +1762.92 q^{51} +4611.29 q^{53} -6537.78i q^{57} -822.630 q^{59} +6917.98i q^{61} -1000.70 q^{63} +4929.83i q^{67} +5077.48i q^{69} +4236.85i q^{71} -8507.29i q^{73} +7147.07 q^{77} +10205.9i q^{79} -7898.82 q^{81} +818.969i q^{83} -5203.25 q^{87} +10451.0 q^{89} -8731.82 q^{91} -5021.46 q^{93} +11691.7i q^{97} -3814.21 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$32 q - 864 q^{9} - 384 q^{11} - 1408 q^{19} - 4416 q^{41} + 4960 q^{49} + 35584 q^{51} + 28032 q^{59} + 20768 q^{81} - 13632 q^{89} - 49152 q^{91} + 5248 q^{99}+O(q^{100})$

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
			$3$	10.2034i	1.13371i	0.823817	+	0.566856i	$0.191840\pi$
−0.823817	+	0.566856i				$0.808160\pi$
$5$	0	0
			$7$	43.3025	0.883724	0.441862	−	0.897083i	$-0.354318\pi$
0.441862	+	0.897083i				$0.354318\pi$
$11$	165.050	1.36405	0.682024	−	0.731329i	$-0.261100\pi$
			0.682024	+	0.731329i	$0.261100\pi$
$13$	−201.647	−1.19318	−0.596589	−	0.802547i	$-0.703478\pi$
			−0.596589	+	0.802547i	$0.703478\pi$
$17$	− 172.778i	− 0.597848i	−0.954277	−	0.298924i	$-0.903372\pi$
			0.954277	−	0.298924i	$-0.0966278\pi$
$19$	−640.745	−1.77492	−0.887458	−	0.460888i	$-0.847531\pi$
			−0.887458	+	0.460888i	$0.847531\pi$
$23$	497.626	0.940692	0.470346	−	0.882482i	$-0.344129\pi$
			0.470346	+	0.882482i	$0.344129\pi$
$29$	509.953i	0.606365i	0.952933	+	0.303182i	$0.0980491\pi$
			−0.952933	+	0.303182i	$0.901951\pi$
$31$	492.136i	0.512108i	0.966662	+	0.256054i	$0.0824225\pi$
			−0.966662	+	0.256054i	$0.917578\pi$
$37$	678.519	0.495631	0.247816	−	0.968807i	$-0.420287\pi$
			0.247816	+	0.968807i	$0.420287\pi$
$41$	−613.848	−0.365168	−0.182584	−	0.983190i	$-0.558446\pi$
			−0.182584	+	0.983190i	$0.558446\pi$
$43$	1392.38i	0.753043i	0.926408	+	0.376522i	$0.122880\pi$
			−0.926408	+	0.376522i	$0.877120\pi$
$47$	965.658	0.437147	0.218574	−	0.975820i	$-0.429860\pi$
			0.218574	+	0.975820i	$0.429860\pi$

(See $a_n$ instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	800.5.e.e.399.10		32
4.3	odd	2		200.5.e.e.99.31		32
5.2	odd	4		160.5.g.a.111.13		16
5.3	odd	4		800.5.g.h.751.3		16
5.4	even	2	inner	800.5.e.e.399.23		32
8.3	odd	2	inner	800.5.e.e.399.24		32
8.5	even	2		200.5.e.e.99.1		32
15.2	even	4		1440.5.g.a.271.14		16
20.3	even	4		200.5.g.h.51.7		16
20.7	even	4		40.5.g.a.11.10	yes	16
20.19	odd	2		200.5.e.e.99.2		32
40.3	even	4		800.5.g.h.751.4		16
40.13	odd	4		200.5.g.h.51.8		16
40.19	odd	2	inner	800.5.e.e.399.9		32
40.27	even	4		160.5.g.a.111.14		16
40.29	even	2		200.5.e.e.99.32		32
40.37	odd	4		40.5.g.a.11.9	✓	16
60.47	odd	4		360.5.g.a.91.7		16
120.77	even	4		360.5.g.a.91.8		16
120.107	odd	4		1440.5.g.a.271.3		16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.5.g.a.11.9	✓	16	40.37	odd	4
40.5.g.a.11.10	yes	16	20.7	even	4
160.5.g.a.111.13		16	5.2	odd	4
160.5.g.a.111.14		16	40.27	even	4
200.5.e.e.99.1		32	8.5	even	2
200.5.e.e.99.2		32	20.19	odd	2
200.5.e.e.99.31		32	4.3	odd	2
200.5.e.e.99.32		32	40.29	even	2
200.5.g.h.51.7		16	20.3	even	4
200.5.g.h.51.8		16	40.13	odd	4
360.5.g.a.91.7		16	60.47	odd	4
360.5.g.a.91.8		16	120.77	even	4
800.5.e.e.399.9		32	40.19	odd	2	inner
800.5.e.e.399.10		32	1.1	even	1	trivial
800.5.e.e.399.23		32	5.4	even	2	inner
800.5.e.e.399.24		32	8.3	odd	2	inner
800.5.g.h.751.3		16	5.3	odd	4
800.5.g.h.751.4		16	40.3	even	4
1440.5.g.a.271.3		16	120.107	odd	4
1440.5.g.a.271.14		16	15.2	even	4