Embedded newform 572.2.s.a.43.1

• Label: 572.2.s.a.43.1
• 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			43.1
Character	\(\chi\)	\(=\)	572.43
Dual form			572.2.s.a.439.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.41333 + 0.0501117i) q^{2} +(1.83041 + 1.05679i) q^{3} +(1.99498 - 0.141648i) q^{4} -1.43229i q^{5} +(-2.63992 - 1.40186i) q^{6} +(1.20221 - 0.694098i) q^{7} +(-2.81245 + 0.300167i) q^{8} +(0.733590 + 1.27062i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q - 2 q^{4} + 68 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.41333	+	0.0501117i	−0.999372	+	0.0354343i
							\(3\)	1.83041	+	1.05679i	1.05679	+	0.610135i	0.924541	−	0.381082i	\(-0.124448\pi\)
0.132244	+	0.991217i	\(0.457782\pi\)
\(5\)		−	1.43229i		−	0.640541i	−0.947326	−	0.320270i	\(-0.896226\pi\)
							0.947326	−	0.320270i	\(-0.103774\pi\)
\(7\)	1.20221	−	0.694098i	0.454394	−	0.262345i	−0.255290	−	0.966865i	\(-0.582171\pi\)
							0.709684	+	0.704520i	\(0.248838\pi\)
\(11\)	−2.37433	−	2.31572i	−0.715887	−	0.698216i
							\(13\)	3.58466	+	0.387551i	0.994206	+	0.107487i
\(17\)	4.51033	−	2.60404i	1.09392	−	0.631572i								0.159299	−	0.987230i	\(-0.449077\pi\)
							0.934616	+	0.355658i	\(0.115743\pi\)
\(19\)	−0.671440	+	0.387656i	−0.154039	+	0.0889345i	−0.575038	−	0.818127i	\(-0.695013\pi\)
							0.420999	+	0.907061i	\(0.361680\pi\)
\(23\)	−0.988183	−	0.570528i	−0.206050	−	0.118963i	0.393424	−	0.919357i	\(-0.371290\pi\)
							−0.599475	+	0.800394i	\(0.704624\pi\)
\(29\)	2.16167	+	1.24804i	0.401412	+	0.231755i	0.687093	−	0.726570i	\(-0.258887\pi\)
							−0.285681	+	0.958325i	\(0.592220\pi\)
\(31\)	−6.07804			−1.09165			−0.545824	−	0.837900i	\(-0.683783\pi\)
							−0.545824	+	0.837900i	\(0.683783\pi\)
\(37\)	3.48741	+	2.01346i	0.573327	+	0.331010i	0.758477	−	0.651700i	\(-0.225944\pi\)
							−0.185150	+	0.982710i	\(0.559277\pi\)
\(41\)	1.22138	−	2.11550i	0.190748	−	0.330385i	−0.754750	−	0.656012i	\(-0.772242\pi\)
							0.945498	+	0.325627i	\(0.105575\pi\)
\(43\)	2.82968	+	4.90115i	0.431523	+	0.747419i	0.997005	−	0.0773415i	\(-0.0246432\pi\)
							−0.565482	+	0.824761i	\(0.691310\pi\)
\(47\)	−1.62915			−0.237636			−0.118818	−	0.992916i	\(-0.537911\pi\)
							−0.118818	+	0.992916i	\(0.537911\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	572.2.s.a.43.1	✓	160
4.3	odd	2	inner	572.2.s.a.43.54	yes	160
11.10	odd	2	inner	572.2.s.a.43.80	yes	160
13.10	even	6	inner	572.2.s.a.439.27	yes	160
44.43	even	2	inner	572.2.s.a.43.27	yes	160
52.23	odd	6	inner	572.2.s.a.439.80	yes	160
143.10	odd	6	inner	572.2.s.a.439.54	yes	160
572.439	even	6	inner	572.2.s.a.439.1	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
572.2.s.a.43.1	✓	160	1.1	even	1	trivial
572.2.s.a.43.27	yes	160	44.43	even	2	inner
572.2.s.a.43.54	yes	160	4.3	odd	2	inner
572.2.s.a.43.80	yes	160	11.10	odd	2	inner
572.2.s.a.439.1	yes	160	572.439	even	6	inner
572.2.s.a.439.27	yes	160	13.10	even	6	inner
572.2.s.a.439.54	yes	160	143.10	odd	6	inner
572.2.s.a.439.80	yes	160	52.23	odd	6	inner