Embedded newform 572.2.s.a.43.10

• Label: 572.2.s.a.43.10
• 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			43.10
Character	\(\chi\)	\(=\)	572.43
Dual form			572.2.s.a.439.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.31800 - 0.512708i) q^{2} +(2.39320 + 1.38172i) q^{3} +(1.47426 + 1.35150i) q^{4} -1.60862i q^{5} +(-2.44583 - 3.04812i) q^{6} +(0.824322 - 0.475923i) q^{7} +(-1.25015 - 2.53715i) q^{8} +(2.31828 + 4.01537i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q - 2 q^{4} + 68 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.31800	−	0.512708i	−0.931968	−	0.362540i
							\(3\)	2.39320	+	1.38172i	1.38172	+	0.797734i	0.992363	−	0.123355i	\(-0.0393653\pi\)
0.389353	+	0.921089i	\(0.372699\pi\)
\(5\)		−	1.60862i		−	0.719397i	−0.933069	−	0.359698i	\(-0.882880\pi\)
							0.933069	−	0.359698i	\(-0.117120\pi\)
\(7\)	0.824322	−	0.475923i	0.311565	−	0.179882i	−0.336062	−	0.941840i	\(-0.609095\pi\)
							0.647626	+	0.761958i	\(0.275762\pi\)
\(11\)	2.08622	+	2.57831i	0.629018	+	0.777391i
							\(13\)	−0.449071	+	3.57748i	−0.124550	+	0.992213i
\(17\)	−3.08043	+	1.77849i	−0.747113	+	0.431346i								−0.824650	−	0.565643i	\(-0.808628\pi\)
							0.0775366	+	0.996990i	\(0.475295\pi\)
\(19\)	5.85339	−	3.37945i	1.34286	−	0.775300i	0.355633	−	0.934626i	\(-0.384265\pi\)
							0.987226	+	0.159326i	\(0.0509321\pi\)
\(23\)	−4.39537	−	2.53767i	−0.916497	−	0.529140i	−0.0339814	−	0.999422i	\(-0.510819\pi\)
							−0.882516	+	0.470283i	\(0.844152\pi\)
\(29\)	0.157432	+	0.0908935i	0.0292344	+	0.0168785i	0.514546	−	0.857463i	\(-0.327960\pi\)
							−0.485312	+	0.874341i	\(0.661294\pi\)
\(31\)	10.5012			1.88608			0.943039	−	0.332681i	\(-0.107953\pi\)
							0.943039	+	0.332681i	\(0.107953\pi\)
\(37\)	−10.2058	−	5.89234i	−1.67783	−	0.968694i	−0.963042	−	0.269351i	\(-0.913191\pi\)
							−0.714786	−	0.699343i	\(-0.753476\pi\)
\(41\)	−1.77565	+	3.07552i	−0.277310	+	0.480316i	−0.970715	−	0.240232i	\(-0.922776\pi\)
							0.693405	+	0.720548i	\(0.256110\pi\)
\(43\)	0.883307	+	1.52993i	0.134703	+	0.233313i	0.925484	−	0.378787i	\(-0.123659\pi\)
							−0.790781	+	0.612099i	\(0.790325\pi\)
\(47\)	−0.738448			−0.107714			−0.0538569	−	0.998549i	\(-0.517151\pi\)
							−0.0538569	+	0.998549i	\(0.517151\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	572.2.s.a.43.10	✓	160
4.3	odd	2	inner	572.2.s.a.43.63	yes	160
11.10	odd	2	inner	572.2.s.a.43.71	yes	160
13.10	even	6	inner	572.2.s.a.439.18	yes	160
44.43	even	2	inner	572.2.s.a.43.18	yes	160
52.23	odd	6	inner	572.2.s.a.439.71	yes	160
143.10	odd	6	inner	572.2.s.a.439.63	yes	160
572.439	even	6	inner	572.2.s.a.439.10	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
572.2.s.a.43.10	✓	160	1.1	even	1	trivial
572.2.s.a.43.18	yes	160	44.43	even	2	inner
572.2.s.a.43.63	yes	160	4.3	odd	2	inner
572.2.s.a.43.71	yes	160	11.10	odd	2	inner
572.2.s.a.439.10	yes	160	572.439	even	6	inner
572.2.s.a.439.18	yes	160	13.10	even	6	inner
572.2.s.a.439.63	yes	160	143.10	odd	6	inner
572.2.s.a.439.71	yes	160	52.23	odd	6	inner