Embedded newform 572.2.t.a.87.11

• Label: 572.2.t.a.87.11
• 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			87.11
Character	\(\chi\)	\(=\)	572.87
Dual form			572.2.t.a.263.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.33705 - 0.460748i) q^{2} +(-0.847243 - 0.489156i) q^{3} +(1.57542 + 1.23209i) q^{4} +1.08222 q^{5} +(0.907431 + 1.04439i) q^{6} +(2.37503 + 4.11368i) q^{7} +(-1.53874 - 2.37324i) q^{8} +(-1.02145 - 1.76921i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 160 q - 2 q^{4} - 16 q^{5} + 68 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.33705	−	0.460748i	−0.945439	−	0.325798i
							\(3\)	−0.847243	−	0.489156i	−0.489156	−	0.282414i	0.235068	−	0.971979i	\(-0.424469\pi\)
−0.724224	+	0.689565i	\(0.757802\pi\)
\(5\)	1.08222			0.483985			0.241993	−	0.970278i	\(-0.422199\pi\)
							0.241993	+	0.970278i	\(0.422199\pi\)
\(7\)	2.37503	+	4.11368i	0.897678	+	1.55482i	0.830455	+	0.557086i	\(0.188081\pi\)
							0.0672233	+	0.997738i	\(0.478586\pi\)
\(11\)	1.33247	+	3.03719i	0.401754	+	0.915748i
							\(13\)	−1.04984	+	3.44932i	−0.291173	+	0.956670i
\(17\)	−2.77472	+	1.60198i	−0.672968	+	0.388538i								−0.797200	−	0.603715i	\(-0.793686\pi\)
							0.124232	+	0.992253i	\(0.460353\pi\)
\(19\)	−1.31893	−	2.28445i	−0.302582	−	0.524088i	0.674138	−	0.738606i	\(-0.264515\pi\)
							−0.976720	+	0.214518i	\(0.931182\pi\)
\(23\)	−1.04372	−	0.602591i	−0.217630	−	0.125649i	0.387222	−	0.921986i	\(-0.373435\pi\)
							−0.604853	+	0.796338i	\(0.706768\pi\)
\(29\)	−7.49285	−	4.32600i	−1.39139	−	0.803318i	−0.397919	−	0.917420i	\(-0.630268\pi\)
							−0.993469	+	0.114102i	\(0.963601\pi\)
\(31\)			5.44201i			0.977414i	0.872448	+	0.488707i	\(0.162531\pi\)
							−0.872448	+	0.488707i	\(0.837469\pi\)
\(37\)	0.654897	−	1.13432i	0.107664	−	0.186480i	−0.807159	−	0.590334i	\(-0.798996\pi\)
							0.914824	+	0.403853i	\(0.132329\pi\)
\(41\)	4.87482	+	2.81448i	0.761319	+	0.439548i	0.829769	−	0.558107i	\(-0.188472\pi\)
							−0.0684503	+	0.997655i	\(0.521805\pi\)
\(43\)	5.40275	+	9.35785i	0.823912	+	1.42706i	0.902748	+	0.430170i	\(0.141547\pi\)
							−0.0788354	+	0.996888i	\(0.525120\pi\)
\(47\)			0.331999i			0.0484271i	0.999707	+	0.0242135i	\(0.00770816\pi\)
							−0.999707	+	0.0242135i	\(0.992292\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	572.2.t.a.87.11	✓	160
4.3	odd	2	inner	572.2.t.a.87.16	yes	160
11.10	odd	2	inner	572.2.t.a.87.70	yes	160
13.3	even	3	inner	572.2.t.a.263.65	yes	160
44.43	even	2	inner	572.2.t.a.87.65	yes	160
52.3	odd	6	inner	572.2.t.a.263.70	yes	160
143.120	odd	6	inner	572.2.t.a.263.16	yes	160
572.263	even	6	inner	572.2.t.a.263.11	yes	160

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
572.2.t.a.87.11	✓	160	1.1	even	1	trivial
572.2.t.a.87.16	yes	160	4.3	odd	2	inner
572.2.t.a.87.65	yes	160	44.43	even	2	inner
572.2.t.a.87.70	yes	160	11.10	odd	2	inner
572.2.t.a.263.11	yes	160	572.263	even	6	inner
572.2.t.a.263.16	yes	160	143.120	odd	6	inner
572.2.t.a.263.65	yes	160	13.3	even	3	inner
572.2.t.a.263.70	yes	160	52.3	odd	6	inner