Embedded newform 572.2.bb.b.51.8

• Label: 572.2.bb.b.51.8
• Level: $572$
• Weight: $2$
• Character: 572.51
• Analytic conductor: $4.567$
• Analytic rank: $0$
• Dimension: $304$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 572 = 2^{2} \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	572.bb (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.56744299562\)
Analytic rank:	\(0\)
Dimension:	\(304\)
Relative dimension:	\(76\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			51.8
Character	\(\chi\)	\(=\)	572.51
Dual form			572.2.bb.b.415.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.33274 - 0.473076i) q^{2} +(-0.481761 - 0.156533i) q^{3} +(1.55240 + 1.26098i) q^{4} +(2.42980 + 3.34433i) q^{5} +(0.568010 + 0.436528i) q^{6} +(4.04744 - 1.31509i) q^{7} +(-1.47241 - 2.41496i) q^{8} +(-2.21946 - 1.61253i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 304 q - 14 q^{4} + 72 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.33274	−	0.473076i	−0.942390	−	0.334516i
							\(3\)	−0.481761	−	0.156533i	−0.278145	−	0.0903747i	0.166623	−	0.986021i	\(-0.446714\pi\)
−0.444768	+	0.895646i	\(0.646714\pi\)
\(5\)	2.42980	+	3.34433i	1.08664	+	1.49563i	0.851994	+	0.523551i	\(0.175393\pi\)
							0.234646	+	0.972081i	\(0.424607\pi\)
\(7\)	4.04744	−	1.31509i	1.52979	−	0.497058i	0.581251	−	0.813724i	\(-0.302563\pi\)
							0.948537	+	0.316666i	\(0.102563\pi\)
\(11\)	−0.00370369	−	3.31662i	−0.00111670	−	0.999999i
							\(13\)	3.20207	−	1.65734i	0.888094	−	0.459663i
\(17\)	−1.30375	−	1.79445i	−0.316205	−	0.435219i								0.621099	−	0.783732i	\(-0.286687\pi\)
							−0.937304	+	0.348514i	\(0.886687\pi\)
\(19\)	4.20184	+	1.36526i	0.963969	+	0.313213i	0.748379	−	0.663271i	\(-0.230832\pi\)
							0.215590	+	0.976484i	\(0.430832\pi\)
\(23\)		−	6.62705i		−	1.38184i	−0.722934	−	0.690918i	\(-0.757207\pi\)
							0.722934	−	0.690918i	\(-0.242793\pi\)
\(29\)	−2.08406	+	0.677151i	−0.387000	+	0.125744i	−0.496053	−	0.868292i	\(-0.665218\pi\)
							0.109054	+	0.994036i	\(0.465218\pi\)
\(31\)	1.82759	+	1.32782i	0.328245	+	0.238484i	0.739685	−	0.672953i	\(-0.234974\pi\)
							−0.411441	+	0.911437i	\(0.634974\pi\)
\(37\)	−3.97897	+	1.29284i	−0.654138	+	0.212542i	−0.617238	−	0.786777i	\(-0.711748\pi\)
							−0.0369001	+	0.999319i	\(0.511748\pi\)
\(41\)	−0.809836	+	2.49242i	−0.126475	+	0.389250i	−0.994167	−	0.107852i	\(-0.965603\pi\)
							0.867692	+	0.497102i	\(0.165603\pi\)
\(43\)	−2.15820			−0.329122			−0.164561	−	0.986367i	\(-0.552621\pi\)
							−0.164561	+	0.986367i	\(0.552621\pi\)
\(47\)	0.843274	−	2.59533i	0.123004	−	0.378568i	−0.870528	−	0.492119i	\(-0.836223\pi\)
							0.993532	+	0.113551i	\(0.0362225\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	572.2.bb.b.51.8	✓	304
4.3	odd	2	inner	572.2.bb.b.51.23	yes	304
11.8	odd	10	inner	572.2.bb.b.415.54	yes	304
13.12	even	2	inner	572.2.bb.b.51.69	yes	304
44.19	even	10	inner	572.2.bb.b.415.69	yes	304
52.51	odd	2	inner	572.2.bb.b.51.54	yes	304
143.129	odd	10	inner	572.2.bb.b.415.23	yes	304
572.415	even	10	inner	572.2.bb.b.415.8	yes	304

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
572.2.bb.b.51.8	✓	304	1.1	even	1	trivial
572.2.bb.b.51.23	yes	304	4.3	odd	2	inner
572.2.bb.b.51.54	yes	304	52.51	odd	2	inner
572.2.bb.b.51.69	yes	304	13.12	even	2	inner
572.2.bb.b.415.8	yes	304	572.415	even	10	inner
572.2.bb.b.415.23	yes	304	143.129	odd	10	inner
572.2.bb.b.415.54	yes	304	11.8	odd	10	inner
572.2.bb.b.415.69	yes	304	44.19	even	10	inner