Embedded newform 572.2.s.a.43.3

• Label: 572.2.s.a.43.3
• 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			43.3
Character	\(\chi\)	\(=\)	572.43
Dual form			572.2.s.a.439.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.40920 - 0.118969i) q^{2} +(-2.07307 - 1.19689i) q^{3} +(1.97169 + 0.335302i) q^{4} +0.562219i q^{5} +(2.77898 + 1.93329i) q^{6} +(-0.112468 + 0.0649336i) q^{7} +(-2.73862 - 0.707079i) q^{8} +(1.36509 + 2.36441i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q - 2 q^{4} + 68 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.40920	−	0.118969i	−0.996455	−	0.0841238i
							\(3\)	−2.07307	−	1.19689i	−1.19689	−	0.691025i	−0.237029	−	0.971502i	\(-0.576174\pi\)
−0.959861	+	0.280478i	\(0.909507\pi\)
\(5\)			0.562219i			0.251432i	0.992066	+	0.125716i	\(0.0401228\pi\)
							−0.992066	+	0.125716i	\(0.959877\pi\)
\(7\)	−0.112468	+	0.0649336i	−0.0425090	+	0.0245426i	−0.521104	−	0.853493i	\(-0.674480\pi\)
							0.478595	+	0.878036i	\(0.341146\pi\)
\(11\)	3.02828	−	1.35260i	0.913060	−	0.407825i
							\(13\)	−3.51999	+	0.780816i	−0.976269	+	0.216559i
\(17\)	−0.0978433	+	0.0564899i	−0.0237305	+	0.0137008i								−0.511818	−	0.859094i	\(-0.671028\pi\)
							0.488088	+	0.872795i	\(0.337695\pi\)
\(19\)	−6.72328	+	3.88168i	−1.54243	+	0.890520i	−0.543740	+	0.839254i	\(0.682992\pi\)
							−0.998685	+	0.0512660i	\(0.983674\pi\)
\(23\)	−6.34375	−	3.66256i	−1.32276	−	0.763698i	−0.338594	−	0.940932i	\(-0.609951\pi\)
							−0.984169	+	0.177235i	\(0.943285\pi\)
\(29\)	6.21458	+	3.58799i	1.15402	+	0.666273i	0.949863	−	0.312665i	\(-0.101222\pi\)
							0.204156	+	0.978938i	\(0.434555\pi\)
\(31\)	−0.573167			−0.102944			−0.0514720	−	0.998674i	\(-0.516391\pi\)
							−0.0514720	+	0.998674i	\(0.516391\pi\)
\(37\)	4.23610	+	2.44571i	0.696410	+	0.402073i	0.806009	−	0.591903i	\(-0.201623\pi\)
							−0.109599	+	0.993976i	\(0.534957\pi\)
\(41\)	0.576345	−	0.998259i	0.0900100	−	0.155902i	−0.817505	−	0.575921i	\(-0.804643\pi\)
							0.907515	+	0.420020i	\(0.137977\pi\)
\(43\)	4.38000	+	7.58638i	0.667943	+	1.15691i	0.978478	+	0.206350i	\(0.0661584\pi\)
							−0.310535	+	0.950562i	\(0.600508\pi\)
\(47\)	5.66927			0.826947			0.413474	−	0.910516i	\(-0.364315\pi\)
							0.413474	+	0.910516i	\(0.364315\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	572.2.s.a.43.3	✓	160
4.3	odd	2	inner	572.2.s.a.43.56	yes	160
11.10	odd	2	inner	572.2.s.a.43.78	yes	160
13.10	even	6	inner	572.2.s.a.439.25	yes	160
44.43	even	2	inner	572.2.s.a.43.25	yes	160
52.23	odd	6	inner	572.2.s.a.439.78	yes	160
143.10	odd	6	inner	572.2.s.a.439.56	yes	160
572.439	even	6	inner	572.2.s.a.439.3	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
572.2.s.a.43.3	✓	160	1.1	even	1	trivial
572.2.s.a.43.25	yes	160	44.43	even	2	inner
572.2.s.a.43.56	yes	160	4.3	odd	2	inner
572.2.s.a.43.78	yes	160	11.10	odd	2	inner
572.2.s.a.439.3	yes	160	572.439	even	6	inner
572.2.s.a.439.25	yes	160	13.10	even	6	inner
572.2.s.a.439.56	yes	160	143.10	odd	6	inner
572.2.s.a.439.78	yes	160	52.23	odd	6	inner