Embedded newform 572.2.bf.a.67.7

• Label: 572.2.bf.a.67.7
• Level: $572$
• Weight: $2$
• Character: 572.67
• Analytic conductor: $4.567$
• Analytic rank: $0$
• Dimension: $280$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 572 = 2^{2} \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	572.bf (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.56744299562\)
Analytic rank:	\(0\)
Dimension:	\(280\)
Relative dimension:	\(70\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			67.7
Character	\(\chi\)	\(=\)	572.67
Dual form			572.2.bf.a.111.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.36347 + 0.375447i) q^{2} +(-0.276166 + 0.159445i) q^{3} +(1.71808 - 1.02382i) q^{4} +(0.625185 - 0.625185i) q^{5} +(0.316680 - 0.321083i) q^{6} +(0.928719 + 3.46603i) q^{7} +(-1.95815 + 2.04099i) q^{8} +(-1.44915 + 2.51001i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 280 q - 4 q^{5} + 12 q^{6} - 12 q^{8} + 140 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(287\)	\(353\)	\(365\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{12}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.36347	+	0.375447i	−0.964116	+	0.265481i
							\(3\)	−0.276166	+	0.159445i	−0.159445	+	0.0920555i	−0.577599	−	0.816320i	\(-0.696010\pi\)
0.418155	+	0.908376i	\(0.362677\pi\)
\(5\)	0.625185	−	0.625185i	0.279591	−	0.279591i	−0.553355	−	0.832946i	\(-0.686653\pi\)
							0.832946	+	0.553355i	\(0.186653\pi\)
\(7\)	0.928719	+	3.46603i	0.351023	+	1.31003i	0.885416	+	0.464799i	\(0.153873\pi\)
							−0.534394	+	0.845236i	\(0.679460\pi\)
\(11\)	0.965926	+	0.258819i	0.291238	+	0.0780369i
							\(13\)	−2.97355	−	2.03912i	−0.824714	−	0.565549i
\(17\)	−0.711607	−	0.410847i	−0.172590	−	0.0996450i								0.411216	−	0.911538i	\(-0.365104\pi\)
							−0.583807	+	0.811893i	\(0.698437\pi\)
\(19\)	−4.56260	+	1.22254i	−1.04673	+	0.280471i	−0.740901	−	0.671615i	\(-0.765601\pi\)
							−0.305831	+	0.952086i	\(0.598934\pi\)
\(23\)	1.99077	+	3.44811i	0.415104	+	0.718981i	0.995439	−	0.0953964i	\(-0.0304119\pi\)
							−0.580335	+	0.814378i	\(0.697079\pi\)
\(29\)	−3.99396	−	6.91775i	−0.741660	−	1.28459i	−0.951739	−	0.306909i	\(-0.900705\pi\)
							0.210079	−	0.977684i	\(-0.432628\pi\)
\(31\)	6.61208	+	6.61208i	1.18756	+	1.18756i	0.977739	+	0.209826i	\(0.0672897\pi\)
							0.209826	+	0.977739i	\(0.432710\pi\)
\(37\)	−0.544502	+	2.03211i	−0.0895156	+	0.334077i	−0.996131	−	0.0878813i	\(-0.971990\pi\)
							0.906615	+	0.421958i	\(0.138657\pi\)
\(41\)	−5.70050	−	1.52745i	−0.890269	−	0.238547i	−0.215437	−	0.976518i	\(-0.569117\pi\)
							−0.674832	+	0.737971i	\(0.735784\pi\)
\(43\)	−1.05038	+	1.81931i	−0.160182	+	0.277442i	−0.934934	−	0.354823i	\(-0.884541\pi\)
							0.774752	+	0.632265i	\(0.217875\pi\)
\(47\)	−4.96519	+	4.96519i	−0.724247	+	0.724247i	−0.969467	−	0.245220i	\(-0.921140\pi\)
							0.245220	+	0.969467i	\(0.421140\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	572.2.bf.a.67.7	✓	280
4.3	odd	2	inner	572.2.bf.a.67.19	yes	280
13.7	odd	12	inner	572.2.bf.a.111.19	yes	280
52.7	even	12	inner	572.2.bf.a.111.7	yes	280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
572.2.bf.a.67.7	✓	280	1.1	even	1	trivial
572.2.bf.a.67.19	yes	280	4.3	odd	2	inner
572.2.bf.a.111.7	yes	280	52.7	even	12	inner
572.2.bf.a.111.19	yes	280	13.7	odd	12	inner