Embedded newform 572.2.bf.a.67.19

• Label: 572.2.bf.a.67.19
• 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			67.19
Character	\(\chi\)	\(=\)	572.67
Dual form			572.2.bf.a.111.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.993073 - 1.00688i) q^{2} +(0.276166 - 0.159445i) q^{3} +(-0.0276135 + 1.99981i) q^{4} +(0.625185 - 0.625185i) q^{5} +(-0.434795 - 0.119726i) q^{6} +(-0.928719 - 3.46603i) q^{7} +(2.04099 - 1.95815i) q^{8} +(-1.44915 + 2.51001i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 280 q - 4 q^{5} + 12 q^{6} - 12 q^{8} + 140 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.993073	−	1.00688i	−0.702208	−	0.711971i
							\(3\)	0.276166	−	0.159445i	0.159445	−	0.0920555i	−0.418155	−	0.908376i	\(-0.637323\pi\)
0.577599	+	0.816320i	\(0.303990\pi\)
\(5\)	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	\(-0.846127\pi\)
							0.534394	−	0.845236i	\(-0.320540\pi\)
\(11\)	−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	0.305831	−	0.952086i	\(-0.401066\pi\)
							0.740901	+	0.671615i	\(0.234399\pi\)
\(23\)	−1.99077	−	3.44811i	−0.415104	−	0.718981i	0.580335	−	0.814378i	\(-0.302921\pi\)
							−0.995439	+	0.0953964i	\(0.969588\pi\)
\(29\)	−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	\(-0.932710\pi\)
							−0.209826	−	0.977739i	\(-0.567290\pi\)
\(37\)	−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	−0.774752	−	0.632265i	\(-0.782125\pi\)
							0.934934	+	0.354823i	\(0.115459\pi\)
\(47\)	4.96519	−	4.96519i	0.724247	−	0.724247i	−0.245220	−	0.969467i	\(-0.578860\pi\)
							0.969467	+	0.245220i	\(0.0788603\pi\)

(See \(a_n\) instead)

Twists

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

    

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