Embedded newform 572.2.s.a.43.8

• Label: 572.2.s.a.43.8
• 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			43.8
Character	\(\chi\)	\(=\)	572.43
Dual form			572.2.s.a.439.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.35805 - 0.394577i) q^{2} +(-1.01294 - 0.584824i) q^{3} +(1.68862 + 1.07171i) q^{4} +1.81934i q^{5} +(1.14487 + 1.19391i) q^{6} +(-2.82002 + 1.62814i) q^{7} +(-1.87036 - 2.12173i) q^{8} +(-0.815963 - 1.41329i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q - 2 q^{4} + 68 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.35805	−	0.394577i	−0.960289	−	0.279008i
							\(3\)	−1.01294	−	0.584824i	−0.584824	−	0.337648i	0.178224	−	0.983990i	\(-0.442965\pi\)
−0.763048	+	0.646342i	\(0.776298\pi\)
\(5\)			1.81934i			0.813632i	0.913510	+	0.406816i	\(0.133361\pi\)
							−0.913510	+	0.406816i	\(0.866639\pi\)
\(7\)	−2.82002	+	1.62814i	−1.06587	+	0.615380i	−0.927050	−	0.374938i	\(-0.877664\pi\)
							−0.138819	+	0.990318i	\(0.544331\pi\)
\(11\)	−1.81576	−	2.77543i	−0.547473	−	0.836824i
							\(13\)	2.91549	−	2.12131i	0.808610	−	0.588344i
\(17\)	−1.80765	+	1.04365i	−0.438419	+	0.253121i								−0.702927	−	0.711262i	\(-0.748124\pi\)
							0.264508	+	0.964384i	\(0.414790\pi\)
\(19\)	2.55509	−	1.47518i	0.586178	−	0.338430i	−0.177407	−	0.984138i	\(-0.556771\pi\)
							0.763585	+	0.645707i	\(0.223437\pi\)
\(23\)	4.64817	+	2.68362i	0.969211	+	0.559574i	0.898996	−	0.437958i	\(-0.144298\pi\)
							0.0702154	+	0.997532i	\(0.477631\pi\)
\(29\)	4.88779	+	2.82197i	0.907641	+	0.524027i	0.879671	−	0.475582i	\(-0.157763\pi\)
							0.0279694	+	0.999609i	\(0.491096\pi\)
\(31\)	8.77762			1.57651			0.788253	−	0.615351i	\(-0.210986\pi\)
							0.788253	+	0.615351i	\(0.210986\pi\)
\(37\)	−4.61336	−	2.66352i	−0.758431	−	0.437881i	0.0703009	−	0.997526i	\(-0.477604\pi\)
							−0.828732	+	0.559645i	\(0.810937\pi\)
\(41\)	1.88749	−	3.26923i	0.294776	−	0.510568i	−0.680156	−	0.733067i	\(-0.738088\pi\)
							0.974933	+	0.222499i	\(0.0714215\pi\)
\(43\)	5.48201	+	9.49512i	0.835999	+	1.44799i	0.893214	+	0.449631i	\(0.148444\pi\)
							−0.0572154	+	0.998362i	\(0.518222\pi\)
\(47\)	−2.60393			−0.379823			−0.189911	−	0.981801i	\(-0.560820\pi\)
							−0.189911	+	0.981801i	\(0.560820\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	572.2.s.a.43.8	✓	160
4.3	odd	2	inner	572.2.s.a.43.62	yes	160
11.10	odd	2	inner	572.2.s.a.43.73	yes	160
13.10	even	6	inner	572.2.s.a.439.19	yes	160
44.43	even	2	inner	572.2.s.a.43.19	yes	160
52.23	odd	6	inner	572.2.s.a.439.73	yes	160
143.10	odd	6	inner	572.2.s.a.439.62	yes	160
572.439	even	6	inner	572.2.s.a.439.8	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
572.2.s.a.43.8	✓	160	1.1	even	1	trivial
572.2.s.a.43.19	yes	160	44.43	even	2	inner
572.2.s.a.43.62	yes	160	4.3	odd	2	inner
572.2.s.a.43.73	yes	160	11.10	odd	2	inner
572.2.s.a.439.8	yes	160	572.439	even	6	inner
572.2.s.a.439.19	yes	160	13.10	even	6	inner
572.2.s.a.439.62	yes	160	143.10	odd	6	inner
572.2.s.a.439.73	yes	160	52.23	odd	6	inner