Embedded newform 572.2.s.a.43.9

• Label: 572.2.s.a.43.9
• Level: $572$
• Weight: $2$
• Character: 572.43
• Analytic conductor: $4.567$
• Analytic rank: $0$
• Dimension: $160$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 572 = 2^{2} \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	572.s (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.56744299562\)
Analytic rank:	\(0\)
Dimension:	\(160\)
Relative dimension:	\(80\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			43.9
Character	\(\chi\)	\(=\)	572.43
Dual form			572.2.s.a.439.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.32499 + 0.494359i) q^{2} +(-0.498527 - 0.287825i) q^{3} +(1.51122 - 1.31004i) q^{4} +1.29030i q^{5} +(0.802835 + 0.134915i) q^{6} +(0.822321 - 0.474767i) q^{7} +(-1.35473 + 2.48289i) q^{8} +(-1.33431 - 2.31110i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q - 2 q^{4} + 68 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/572\mathbb{Z}\right)^\times\).

\(n\)	\(287\)	\(353\)	\(365\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{6}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.32499	+	0.494359i	−0.936912	+	0.349564i
							\(3\)	−0.498527	−	0.287825i	−0.287825	−	0.166176i	0.349136	−	0.937072i	\(-0.386475\pi\)
−0.636961	+	0.770896i	\(0.719809\pi\)
\(5\)			1.29030i			0.577041i	0.957474	+	0.288521i	\(0.0931633\pi\)
							−0.957474	+	0.288521i	\(0.906837\pi\)
\(7\)	0.822321	−	0.474767i	0.310808	−	0.179445i	−0.336480	−	0.941691i	\(-0.609236\pi\)
							0.647288	+	0.762246i	\(0.275903\pi\)
\(11\)	−1.19106	−	3.09538i	−0.359119	−	0.933292i
							\(13\)	−2.73142	+	2.35358i	−0.757560	+	0.652765i
\(17\)	−6.12089	+	3.53390i	−1.48453	+	0.857096i								−0.999845	−	0.0175906i	\(-0.994400\pi\)
							−0.484689	+	0.874687i	\(0.661067\pi\)
\(19\)	4.10994	−	2.37288i	0.942886	−	0.544375i	0.0520222	−	0.998646i	\(-0.483433\pi\)
							0.890864	+	0.454270i	\(0.150100\pi\)
\(23\)	−0.314446	−	0.181546i	−0.0655665	−	0.0378549i	0.466858	−	0.884332i	\(-0.345386\pi\)
							−0.532425	+	0.846477i	\(0.678719\pi\)
\(29\)	−4.54470	−	2.62389i	−0.843930	−	0.487243i	0.0146680	−	0.999892i	\(-0.495331\pi\)
							−0.858598	+	0.512649i	\(0.828664\pi\)
\(31\)	−6.62184			−1.18932			−0.594659	−	0.803978i	\(-0.702713\pi\)
							−0.594659	+	0.803978i	\(0.702713\pi\)
\(37\)	−3.35551	−	1.93731i	−0.551643	−	0.318491i	0.198141	−	0.980173i	\(-0.436509\pi\)
							−0.749784	+	0.661682i	\(0.769843\pi\)
\(41\)	3.19323	−	5.53083i	0.498698	−	0.863771i	−0.501301	−	0.865273i	\(-0.667145\pi\)
							0.999999	+	0.00150235i	\(0.000478212\pi\)
\(43\)	−5.12563	−	8.87784i	−0.781651	−	1.35386i	−0.930980	−	0.365071i	\(-0.881045\pi\)
							0.149329	−	0.988788i	\(-0.452289\pi\)
\(47\)	−7.25809			−1.05870			−0.529351	−	0.848403i	\(-0.677564\pi\)
							−0.529351	+	0.848403i	\(0.677564\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	572.2.s.a.43.9	✓	160
4.3	odd	2	inner	572.2.s.a.43.44	yes	160
11.10	odd	2	inner	572.2.s.a.43.72	yes	160
13.10	even	6	inner	572.2.s.a.439.37	yes	160
44.43	even	2	inner	572.2.s.a.43.37	yes	160
52.23	odd	6	inner	572.2.s.a.439.72	yes	160
143.10	odd	6	inner	572.2.s.a.439.44	yes	160
572.439	even	6	inner	572.2.s.a.439.9	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
572.2.s.a.43.9	✓	160	1.1	even	1	trivial
572.2.s.a.43.37	yes	160	44.43	even	2	inner
572.2.s.a.43.44	yes	160	4.3	odd	2	inner
572.2.s.a.43.72	yes	160	11.10	odd	2	inner
572.2.s.a.439.9	yes	160	572.439	even	6	inner
572.2.s.a.439.37	yes	160	13.10	even	6	inner
572.2.s.a.439.44	yes	160	143.10	odd	6	inner
572.2.s.a.439.72	yes	160	52.23	odd	6	inner