Embedded newform 572.2.t.a.87.3

• Label: 572.2.t.a.87.3
• 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			87.3
Character	\(\chi\)	\(=\)	572.87
Dual form			572.2.t.a.263.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.39957 - 0.203000i) q^{2} +(-2.41412 - 1.39379i) q^{3} +(1.91758 + 0.568224i) q^{4} +2.71349 q^{5} +(3.09579 + 2.44077i) q^{6} +(-1.13343 - 1.96316i) q^{7} +(-2.56844 - 1.18454i) q^{8} +(2.38532 + 4.13149i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q - 2 q^{4} - 16 q^{5} + 68 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.39957	−	0.203000i	−0.989644	−	0.143542i
							\(3\)	−2.41412	−	1.39379i	−1.39379	−	0.804707i	−0.400060	−	0.916489i	\(-0.631011\pi\)
−0.993733	+	0.111782i	\(0.964344\pi\)
\(5\)	2.71349			1.21351			0.606754	−	0.794890i	\(-0.292471\pi\)
							0.606754	+	0.794890i	\(0.292471\pi\)
\(7\)	−1.13343	−	1.96316i	−0.428397	−	0.742005i	0.568334	−	0.822798i	\(-0.307588\pi\)
							−0.996731	+	0.0807928i	\(0.974255\pi\)
\(11\)	3.11347	−	1.14295i	0.938745	−	0.344611i
							\(13\)	3.60351	+	0.121185i	0.999435	+	0.0336105i
\(17\)	−1.99873	+	1.15397i	−0.484764	+	0.279878i								−0.722400	−	0.691476i	\(-0.756961\pi\)
							0.237636	+	0.971354i	\(0.423628\pi\)
\(19\)	2.60607	+	4.51385i	0.597874	+	1.03555i	0.993134	+	0.116979i	\(0.0373210\pi\)
							−0.395260	+	0.918569i	\(0.629346\pi\)
\(23\)	6.99460	+	4.03834i	1.45848	+	0.842051i	0.998937	−	0.0461061i	\(-0.0146812\pi\)
							0.459539	+	0.888157i	\(0.348015\pi\)
\(29\)	−2.90271	−	1.67588i	−0.539019	−	0.311203i	0.205662	−	0.978623i	\(-0.434065\pi\)
							−0.744681	+	0.667420i	\(0.767399\pi\)
\(31\)		−	1.40204i		−	0.251814i	−0.992042	−	0.125907i	\(-0.959816\pi\)
							0.992042	−	0.125907i	\(-0.0401842\pi\)
\(37\)	4.79750	−	8.30952i	0.788705	−	1.36608i	−0.138056	−	0.990424i	\(-0.544085\pi\)
							0.926761	−	0.375652i	\(-0.122581\pi\)
\(41\)	3.92987	+	2.26891i	0.613743	+	0.354345i	0.774429	−	0.632661i	\(-0.218037\pi\)
							−0.160686	+	0.987006i	\(0.551371\pi\)
\(43\)	0.935175	+	1.61977i	0.142613	+	0.247013i	0.928480	−	0.371383i	\(-0.121116\pi\)
							−0.785867	+	0.618396i	\(0.787783\pi\)
\(47\)		−	11.1071i		−	1.62013i	−0.586338	−	0.810067i	\(-0.699431\pi\)
							0.586338	−	0.810067i	\(-0.300569\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	572.2.t.a.87.3	✓	160
4.3	odd	2	inner	572.2.t.a.87.25	yes	160
11.10	odd	2	inner	572.2.t.a.87.78	yes	160
13.3	even	3	inner	572.2.t.a.263.56	yes	160
44.43	even	2	inner	572.2.t.a.87.56	yes	160
52.3	odd	6	inner	572.2.t.a.263.78	yes	160
143.120	odd	6	inner	572.2.t.a.263.25	yes	160
572.263	even	6	inner	572.2.t.a.263.3	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
572.2.t.a.87.3	✓	160	1.1	even	1	trivial
572.2.t.a.87.25	yes	160	4.3	odd	2	inner
572.2.t.a.87.56	yes	160	44.43	even	2	inner
572.2.t.a.87.78	yes	160	11.10	odd	2	inner
572.2.t.a.263.3	yes	160	572.263	even	6	inner
572.2.t.a.263.25	yes	160	143.120	odd	6	inner
572.2.t.a.263.56	yes	160	13.3	even	3	inner
572.2.t.a.263.78	yes	160	52.3	odd	6	inner