Embedded newform 572.2.bn.a.95.10

• Label: 572.2.bn.a.95.10
• Level: $572$
• Weight: $2$
• Character: 572.95
• Analytic conductor: $4.567$
• Analytic rank: $0$
• Dimension: $640$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 572 = 2^{2} \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	572.bn (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.56744299562\)
Analytic rank:	\(0\)
Dimension:	\(640\)
Relative dimension:	\(80\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

Embedding invariants

Embedding label			95.10
Character	\(\chi\)	\(=\)	572.95
Dual form			572.2.bn.a.283.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.32607 + 0.491461i) q^{2} +(1.23162 - 1.10895i) q^{3} +(1.51693 - 1.30343i) q^{4} +(0.467022 + 0.642801i) q^{5} +(-1.08820 + 2.07584i) q^{6} +(-3.46433 - 3.11929i) q^{7} +(-1.37098 + 2.47395i) q^{8} +(-0.0264818 + 0.251958i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q - 15 q^{2} - 3 q^{4} - 15 q^{6} - 78 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/572\mathbb{Z}\right)^\times\).

\(n\)	\(287\)	\(353\)	\(365\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{7}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.32607	+	0.491461i	−0.937674	+	0.347516i
							\(3\)	1.23162	−	1.10895i	0.711074	−	0.640254i	−0.232047	−	0.972705i	\(-0.574542\pi\)
0.943121	+	0.332451i	\(0.107876\pi\)
\(5\)	0.467022	+	0.642801i	0.208859	+	0.287469i	0.900576	−	0.434700i	\(-0.143145\pi\)
							−0.691717	+	0.722169i	\(0.743145\pi\)
\(7\)	−3.46433	−	3.11929i	−1.30939	−	1.17898i	−0.971297	−	0.237870i	\(-0.923551\pi\)
							−0.338095	−	0.941112i	\(-0.609782\pi\)
\(11\)	−1.80498	+	2.78245i	−0.544223	+	0.838940i
							\(13\)	2.62434	−	2.47242i	0.727861	−	0.685725i
\(17\)	2.86786	−	6.44131i	0.695557	−	1.56225i								−0.125942	−	0.992038i	\(-0.540195\pi\)
							0.821499	−	0.570209i	\(-0.193138\pi\)
\(19\)	−1.32072	−	6.21351i	−0.302995	−	1.42548i	−0.821405	−	0.570346i	\(-0.806809\pi\)
							0.518410	−	0.855132i	\(-0.326524\pi\)
\(23\)	−3.12267	−	1.80287i	−0.651122	−	0.375925i	0.137764	−	0.990465i	\(-0.456008\pi\)
							−0.788886	+	0.614540i	\(0.789342\pi\)
\(29\)	−0.733459	+	3.45065i	−0.136200	+	0.640770i	0.856090	+	0.516828i	\(0.172887\pi\)
							−0.992289	+	0.123942i	\(0.960446\pi\)
\(31\)	−4.33166	−	3.14713i	−0.777989	−	0.565242i	0.126386	−	0.991981i	\(-0.459662\pi\)
							−0.904375	+	0.426739i	\(0.859662\pi\)
\(37\)	−0.663412	+	3.12111i	−0.109064	+	0.513107i	0.889371	+	0.457186i	\(0.151143\pi\)
							−0.998435	+	0.0559207i	\(0.982191\pi\)
\(41\)	−5.12144	−	5.68794i	−0.799835	−	0.888306i	0.195895	−	0.980625i	\(-0.437239\pi\)
							−0.995730	+	0.0923185i	\(0.970572\pi\)
\(43\)	−0.251423	−	0.435478i	−0.0383417	−	0.0664098i	0.846218	−	0.532837i	\(-0.178874\pi\)
							−0.884559	+	0.466427i	\(0.845541\pi\)
\(47\)	1.22687	−	3.77593i	0.178958	−	0.550775i	−0.820834	−	0.571166i	\(-0.806491\pi\)
							0.999792	+	0.0203910i	\(0.00649111\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	572.2.bn.a.95.10	✓	640
4.3	odd	2	inner	572.2.bn.a.95.13	yes	640
11.8	odd	10	inner	572.2.bn.a.459.40	yes	640
13.10	even	6	inner	572.2.bn.a.491.36	yes	640
44.19	even	10	inner	572.2.bn.a.459.36	yes	640
52.23	odd	6	inner	572.2.bn.a.491.40	yes	640
143.140	odd	30	inner	572.2.bn.a.283.13	yes	640
572.283	even	30	inner	572.2.bn.a.283.10	yes	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
572.2.bn.a.95.10	✓	640	1.1	even	1	trivial
572.2.bn.a.95.13	yes	640	4.3	odd	2	inner
572.2.bn.a.283.10	yes	640	572.283	even	30	inner
572.2.bn.a.283.13	yes	640	143.140	odd	30	inner
572.2.bn.a.459.36	yes	640	44.19	even	10	inner
572.2.bn.a.459.40	yes	640	11.8	odd	10	inner
572.2.bn.a.491.36	yes	640	13.10	even	6	inner
572.2.bn.a.491.40	yes	640	52.23	odd	6	inner