Embedded newform 572.2.t.a.87.7

• Label: 572.2.t.a.87.7
• Level: $572$
• Weight: $2$
• Character: 572.87
• 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			87.7
Character	\(\chi\)	\(=\)	572.87
Dual form			572.2.t.a.263.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.35748 + 0.396562i) q^{2} +(-0.410459 - 0.236978i) q^{3} +(1.68548 - 1.07664i) q^{4} -1.00051 q^{5} +(0.651164 + 0.158920i) q^{6} +(-1.25701 - 2.17721i) q^{7} +(-1.86104 + 2.12991i) q^{8} +(-1.38768 - 2.40354i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q - 2 q^{4} - 16 q^{5} + 68 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(287\)	\(353\)	\(365\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{2}{3}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.35748	+	0.396562i	−0.959880	+	0.280411i
							\(3\)	−0.410459	−	0.236978i	−0.236978	−	0.136820i	0.376809	−	0.926291i	\(-0.377021\pi\)
−0.613787	+	0.789472i	\(0.710355\pi\)
\(5\)	−1.00051			−0.447442			−0.223721	−	0.974653i	\(-0.571821\pi\)
							−0.223721	+	0.974653i	\(0.571821\pi\)
\(7\)	−1.25701	−	2.17721i	−0.475106	−	0.822908i	0.524487	−	0.851418i	\(-0.324257\pi\)
							−0.999594	+	0.0285102i	\(0.990924\pi\)
\(11\)	2.85186	+	1.69319i	0.859869	+	0.510515i
							\(13\)	−3.21305	+	1.63594i	−0.891140	+	0.453728i
\(17\)	−3.75926	+	2.17041i	−0.911755	+	0.526402i								−0.880996	−	0.473125i	\(-0.843126\pi\)
							−0.0307599	+	0.999527i	\(0.509793\pi\)
\(19\)	1.85055	+	3.20525i	0.424546	+	0.735335i	0.996378	−	0.0850357i	\(-0.0271004\pi\)
							−0.571832	+	0.820371i	\(0.693767\pi\)
\(23\)	3.63473	+	2.09851i	0.757894	+	0.437570i	0.828539	−	0.559931i	\(-0.189172\pi\)
							−0.0706449	+	0.997502i	\(0.522506\pi\)
\(29\)	3.13181	+	1.80815i	0.581563	+	0.335766i	0.761754	−	0.647866i	\(-0.224338\pi\)
							−0.180191	+	0.983632i	\(0.557672\pi\)
\(31\)		−	2.13964i		−	0.384290i	−0.981367	−	0.192145i	\(-0.938456\pi\)
							0.981367	−	0.192145i	\(-0.0615444\pi\)
\(37\)	−5.60691	+	9.71144i	−0.921770	+	1.59655i	−0.125094	+	0.992145i	\(0.539923\pi\)
							−0.796676	+	0.604407i	\(0.793410\pi\)
\(41\)	2.38352	+	1.37612i	0.372243	+	0.214914i	0.674438	−	0.738332i	\(-0.264386\pi\)
							−0.302195	+	0.953246i	\(0.597719\pi\)
\(43\)	−3.15711	−	5.46827i	−0.481454	−	0.833903i	0.518319	−	0.855187i	\(-0.326558\pi\)
							−0.999773	+	0.0212843i	\(0.993224\pi\)
\(47\)			8.55870i			1.24842i	0.781259	+	0.624208i	\(0.214578\pi\)
							−0.781259	+	0.624208i	\(0.785422\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	572.2.t.a.87.7	✓	160
4.3	odd	2	inner	572.2.t.a.87.36	yes	160
11.10	odd	2	inner	572.2.t.a.87.74	yes	160
13.3	even	3	inner	572.2.t.a.263.45	yes	160
44.43	even	2	inner	572.2.t.a.87.45	yes	160
52.3	odd	6	inner	572.2.t.a.263.74	yes	160
143.120	odd	6	inner	572.2.t.a.263.36	yes	160
572.263	even	6	inner	572.2.t.a.263.7	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
572.2.t.a.87.7	✓	160	1.1	even	1	trivial
572.2.t.a.87.36	yes	160	4.3	odd	2	inner
572.2.t.a.87.45	yes	160	44.43	even	2	inner
572.2.t.a.87.74	yes	160	11.10	odd	2	inner
572.2.t.a.263.7	yes	160	572.263	even	6	inner
572.2.t.a.263.36	yes	160	143.120	odd	6	inner
572.2.t.a.263.45	yes	160	13.3	even	3	inner
572.2.t.a.263.74	yes	160	52.3	odd	6	inner