Embedded newform 572.2.t.a.263.30

• Label: 572.2.t.a.263.30
• Level: $572$
• Weight: $2$
• Character: 572.263
• 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.t (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			263.30
Character	\(\chi\)	\(=\)	572.263
Dual form			572.2.t.a.87.30

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.499262 + 1.32315i) q^{2} +(-2.08234 + 1.20224i) q^{3} +(-1.50148 - 1.32120i) q^{4} -2.36335 q^{5} +(-0.551117 - 3.35549i) q^{6} +(1.42248 - 2.46380i) q^{7} +(2.49778 - 1.32706i) q^{8} +(1.39077 - 2.40888i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q - 2 q^{4} - 16 q^{5} + 68 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.499262	+	1.32315i	−0.353031	+	0.935612i
							\(3\)	−2.08234	+	1.20224i	−1.20224	+	0.694114i	−0.961053	−	0.276364i	\(-0.910870\pi\)
−0.241188	+	0.970478i	\(0.577537\pi\)
\(5\)	−2.36335			−1.05692			−0.528460	−	0.848958i	\(-0.677230\pi\)
							−0.528460	+	0.848958i	\(0.677230\pi\)
\(7\)	1.42248	−	2.46380i	0.537646	−	0.931230i	−0.461384	−	0.887200i	\(-0.652647\pi\)
							0.999030	−	0.0440296i	\(-0.0140196\pi\)
\(11\)	3.21380	−	0.819455i	0.968996	−	0.247075i
							\(13\)	−2.34026	+	2.74285i	−0.649070	+	0.760729i
\(17\)	−1.53626	−	0.886961i	−0.372598	−	0.215120i								0.301995	−	0.953310i	\(-0.402348\pi\)
							−0.674593	+	0.738190i	\(0.735681\pi\)
\(19\)	−0.0598616	+	0.103683i	−0.0137332	+	0.0237866i	−0.872810	−	0.488060i	\(-0.837705\pi\)
							0.859077	+	0.511846i	\(0.171038\pi\)
\(23\)	5.93594	−	3.42711i	1.23773	−	0.714603i	0.269099	−	0.963113i	\(-0.413274\pi\)
							0.968629	+	0.248510i	\(0.0799408\pi\)
\(29\)	−2.88768	+	1.66721i	−0.536229	+	0.309592i	−0.743549	−	0.668681i	\(-0.766859\pi\)
							0.207320	+	0.978273i	\(0.433526\pi\)
\(31\)			6.06160i			1.08870i	0.838860	+	0.544348i	\(0.183223\pi\)
							−0.838860	+	0.544348i	\(0.816777\pi\)
\(37\)	2.26363	+	3.92072i	0.372139	+	0.644563i	0.989894	−	0.141807i	\(-0.0452912\pi\)
							−0.617756	+	0.786370i	\(0.711958\pi\)
\(41\)	7.34222	−	4.23903i	1.14666	−	0.662026i	0.198591	−	0.980082i	\(-0.436364\pi\)
							0.948072	+	0.318057i	\(0.103030\pi\)
\(43\)	−4.01118	+	6.94756i	−0.611699	+	1.05949i	0.379255	+	0.925292i	\(0.376180\pi\)
							−0.990954	+	0.134201i	\(0.957153\pi\)
\(47\)		−	2.87708i		−	0.419665i	−0.977737	−	0.209833i	\(-0.932708\pi\)
							0.977737	−	0.209833i	\(-0.0672919\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	572.2.t.a.263.30	yes	160
4.3	odd	2	inner	572.2.t.a.263.5	yes	160
11.10	odd	2	inner	572.2.t.a.263.51	yes	160
13.9	even	3	inner	572.2.t.a.87.76	yes	160
44.43	even	2	inner	572.2.t.a.263.76	yes	160
52.35	odd	6	inner	572.2.t.a.87.51	yes	160
143.87	odd	6	inner	572.2.t.a.87.5	✓	160
572.87	even	6	inner	572.2.t.a.87.30	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
572.2.t.a.87.5	✓	160	143.87	odd	6	inner
572.2.t.a.87.30	yes	160	572.87	even	6	inner
572.2.t.a.87.51	yes	160	52.35	odd	6	inner
572.2.t.a.87.76	yes	160	13.9	even	3	inner
572.2.t.a.263.5	yes	160	4.3	odd	2	inner
572.2.t.a.263.30	yes	160	1.1	even	1	trivial
572.2.t.a.263.51	yes	160	11.10	odd	2	inner
572.2.t.a.263.76	yes	160	44.43	even	2	inner