Embedded newform 572.2.bk.a.31.16

• Label: 572.2.bk.a.31.16
• Level: $572$
• Weight: $2$
• Character: 572.31
• 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.bk (of order \(20\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.56744299562\)
Analytic rank:	\(0\)
Dimension:	\(640\)
Relative dimension:	\(80\) over \(\Q(\zeta_{20})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{20}]$

Embedding invariants

Embedding label			31.16
Character	\(\chi\)	\(=\)	572.31
Dual form			572.2.bk.a.203.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.19169 - 0.761498i) q^{2} +(1.13752 + 1.56566i) q^{3} +(0.840242 + 1.81494i) q^{4} +(-1.23921 - 0.631408i) q^{5} +(-0.163321 - 2.73200i) q^{6} +(3.50929 - 0.555817i) q^{7} +(0.380764 - 2.80268i) q^{8} +(-0.230299 + 0.708787i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q - 6 q^{2} - 12 q^{5} - 14 q^{6} - 6 q^{8} + 120 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(287\)	\(353\)	\(365\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{3}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.19169	−	0.761498i	−0.842651	−	0.538460i
							\(3\)	1.13752	+	1.56566i	0.656749	+	0.903937i	0.999368	−	0.0355372i	\(-0.0113142\pi\)
−0.342620	+	0.939474i	\(0.611314\pi\)
\(5\)	−1.23921	−	0.631408i	−0.554190	−	0.282374i	0.154383	−	0.988011i	\(-0.450661\pi\)
							−0.708574	+	0.705637i	\(0.750661\pi\)
\(7\)	3.50929	−	0.555817i	1.32639	−	0.210079i	0.547269	−	0.836957i	\(-0.315667\pi\)
							0.779118	+	0.626878i	\(0.215667\pi\)
\(11\)	−0.254618	−	3.30684i	−0.0767702	−	0.997049i
							\(13\)	2.37399	−	2.71370i	0.658425	−	0.752646i
\(17\)	−0.146340	+	0.0475489i	−0.0354928	+	0.0115323i								−0.326710	−	0.945125i	\(-0.605940\pi\)
							0.291217	+	0.956657i	\(0.405940\pi\)
\(19\)	1.36493	+	0.216184i	0.313137	+	0.0495960i	0.311025	−	0.950402i	\(-0.399328\pi\)
							0.00211181	+	0.999998i	\(0.499328\pi\)
\(23\)	−5.46050			−1.13859			−0.569297	−	0.822132i	\(-0.692785\pi\)
							−0.569297	+	0.822132i	\(0.692785\pi\)
\(29\)	0.877604	+	0.637617i	0.162967	+	0.118402i	0.666280	−	0.745702i	\(-0.267886\pi\)
							−0.503313	+	0.864104i	\(0.667886\pi\)
\(31\)	7.27009	−	3.70429i	1.30575	−	0.665311i	0.343927	−	0.938996i	\(-0.388243\pi\)
							0.961819	+	0.273686i	\(0.0882428\pi\)
\(37\)	1.05306	+	6.64873i	0.173121	+	1.09304i	0.909265	+	0.416219i	\(0.136645\pi\)
							−0.736143	+	0.676826i	\(0.763355\pi\)
\(41\)	−1.06297	+	6.71134i	−0.166008	+	1.04814i	0.754182	+	0.656665i	\(0.228034\pi\)
							−0.920191	+	0.391471i	\(0.871966\pi\)
\(43\)	1.79635			0.273941			0.136971	−	0.990575i	\(-0.456263\pi\)
							0.136971	+	0.990575i	\(0.456263\pi\)
\(47\)	−0.596729	+	3.76760i	−0.0870419	+	0.549561i	0.905175	+	0.425039i	\(0.139740\pi\)
							−0.992217	+	0.124522i	\(0.960260\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	572.2.bk.a.31.16	✓	640
4.3	odd	2	inner	572.2.bk.a.31.41	yes	640
11.5	even	5	inner	572.2.bk.a.291.79	yes	640
13.8	odd	4	inner	572.2.bk.a.515.55	yes	640
44.27	odd	10	inner	572.2.bk.a.291.55	yes	640
52.47	even	4	inner	572.2.bk.a.515.79	yes	640
143.60	odd	20	inner	572.2.bk.a.203.41	yes	640
572.203	even	20	inner	572.2.bk.a.203.16	yes	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
572.2.bk.a.31.16	✓	640	1.1	even	1	trivial
572.2.bk.a.31.41	yes	640	4.3	odd	2	inner
572.2.bk.a.203.16	yes	640	572.203	even	20	inner
572.2.bk.a.203.41	yes	640	143.60	odd	20	inner
572.2.bk.a.291.55	yes	640	44.27	odd	10	inner
572.2.bk.a.291.79	yes	640	11.5	even	5	inner
572.2.bk.a.515.55	yes	640	13.8	odd	4	inner
572.2.bk.a.515.79	yes	640	52.47	even	4	inner