Embedded newform 572.2.s.a.43.4

• Label: 572.2.s.a.43.4
• 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			43.4
Character	\(\chi\)	\(=\)	572.43
Dual form			572.2.s.a.439.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.40794 - 0.133100i) q^{2} +(1.15080 + 0.664413i) q^{3} +(1.96457 + 0.374794i) q^{4} +4.34420i q^{5} +(-1.53182 - 1.08862i) q^{6} +(-0.561519 + 0.324193i) q^{7} +(-2.71610 - 0.789171i) q^{8} +(-0.617110 - 1.06887i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q - 2 q^{4} + 68 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.40794	−	0.133100i	−0.995561	−	0.0941162i
							\(3\)	1.15080	+	0.664413i	0.664413	+	0.383599i	0.793956	−	0.607975i	\(-0.208018\pi\)
−0.129543	+	0.991574i	\(0.541351\pi\)
\(5\)			4.34420i			1.94278i	0.237482	+	0.971392i	\(0.423678\pi\)
							−0.237482	+	0.971392i	\(0.576322\pi\)
\(7\)	−0.561519	+	0.324193i	−0.212234	+	0.122533i	−0.602349	−	0.798233i	\(-0.705769\pi\)
							0.390115	+	0.920766i	\(0.372435\pi\)
\(11\)	3.29882	+	0.343203i	0.994632	+	0.103480i
							\(13\)	0.685537	+	3.53978i	0.190134	+	0.981758i
\(17\)	2.28711	−	1.32046i	0.554705	−	0.320259i								−0.196312	−	0.980541i	\(-0.562897\pi\)
							0.751018	+	0.660282i	\(0.229563\pi\)
\(19\)	−2.34520	+	1.35400i	−0.538027	+	0.310630i	−0.744279	−	0.667869i	\(-0.767207\pi\)
							0.206252	+	0.978499i	\(0.433873\pi\)
\(23\)	4.93889	+	2.85147i	1.02983	+	0.594573i	0.916936	−	0.399034i	\(-0.130655\pi\)
							0.112894	+	0.993607i	\(0.463988\pi\)
\(29\)	0.378658	+	0.218618i	0.0703150	+	0.0405964i	0.534745	−	0.845013i	\(-0.320408\pi\)
							−0.464430	+	0.885610i	\(0.653741\pi\)
\(31\)	−5.69196			−1.02231			−0.511154	−	0.859489i	\(-0.670782\pi\)
							−0.511154	+	0.859489i	\(0.670782\pi\)
\(37\)	−2.06781	−	1.19385i	−0.339946	−	0.196268i	0.320302	−	0.947316i	\(-0.396216\pi\)
							−0.660248	+	0.751047i	\(0.729549\pi\)
\(41\)	−4.66151	+	8.07397i	−0.728005	+	1.26094i	0.229720	+	0.973257i	\(0.426219\pi\)
							−0.957725	+	0.287685i	\(0.907114\pi\)
\(43\)	−4.64105	−	8.03853i	−0.707753	−	1.22586i	−0.965689	−	0.259703i	\(-0.916375\pi\)
							0.257935	−	0.966162i	\(-0.416958\pi\)
\(47\)	9.75831			1.42340			0.711698	−	0.702485i	\(-0.247926\pi\)
							0.711698	+	0.702485i	\(0.247926\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	572.2.s.a.43.4	✓	160
4.3	odd	2	inner	572.2.s.a.43.58	yes	160
11.10	odd	2	inner	572.2.s.a.43.77	yes	160
13.10	even	6	inner	572.2.s.a.439.23	yes	160
44.43	even	2	inner	572.2.s.a.43.23	yes	160
52.23	odd	6	inner	572.2.s.a.439.77	yes	160
143.10	odd	6	inner	572.2.s.a.439.58	yes	160
572.439	even	6	inner	572.2.s.a.439.4	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
572.2.s.a.43.4	✓	160	1.1	even	1	trivial
572.2.s.a.43.23	yes	160	44.43	even	2	inner
572.2.s.a.43.58	yes	160	4.3	odd	2	inner
572.2.s.a.43.77	yes	160	11.10	odd	2	inner
572.2.s.a.439.4	yes	160	572.439	even	6	inner
572.2.s.a.439.23	yes	160	13.10	even	6	inner
572.2.s.a.439.58	yes	160	143.10	odd	6	inner
572.2.s.a.439.77	yes	160	52.23	odd	6	inner