Embedded newform 471.2.l.b.50.16

• Label: 471.2.l.b.50.16
• Level: $471$
• Weight: $2$
• Character: 471.50
• Analytic conductor: $3.761$
• Analytic rank: $0$
• Dimension: $200$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 471 = 3 \cdot 157 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	471.l (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.76095393520\)
Analytic rank:	\(0\)
Dimension:	\(200\)
Relative dimension:	\(50\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			50.16
Character	\(\chi\)	\(=\)	471.50
Dual form			471.2.l.b.179.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.850678 + 0.850678i) q^{2} +(-1.36219 - 1.06978i) q^{3} +0.552695i q^{4} +(0.0315701 + 0.117821i) q^{5} +(2.06883 - 0.248744i) q^{6} +(0.641782 - 0.641782i) q^{7} +(-2.17152 - 2.17152i) q^{8} +(0.711127 + 2.91450i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 200 q - 6 q^{3} - 10 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/471\mathbb{Z}\right)^\times\).

\(n\)	\(158\)	\(319\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{11}{12}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.850678	+	0.850678i	−0.601520	+	0.601520i	−0.940716	−	0.339196i	\(-0.889845\pi\)
							0.339196	+	0.940716i	\(0.389845\pi\)
\(3\)	−1.36219	−	1.06978i	−0.786461	−	0.617640i
							\(5\)	0.0315701	+	0.117821i	0.0141186	+	0.0526912i	0.972626	−	0.232377i	\(-0.0746504\pi\)
−0.958507	+	0.285068i	\(0.907984\pi\)
\(7\)	0.641782	−	0.641782i	0.242571	−	0.242571i	−0.575342	−	0.817913i	\(-0.695131\pi\)
							0.817913	+	0.575342i	\(0.195131\pi\)
\(11\)	−1.92510	−	1.11146i	−0.580440	−	0.335117i	0.180869	−	0.983507i	\(-0.442109\pi\)
							−0.761308	+	0.648390i	\(0.775442\pi\)
\(13\)	0.220101	−	0.127075i	0.0610449	−	0.0352443i	−0.469167	−	0.883110i	\(-0.655446\pi\)
							0.530212	+	0.847865i	\(0.322112\pi\)
\(17\)	0.364781	+	0.210606i	0.0884724	+	0.0510795i	0.543583	−	0.839355i	\(-0.317067\pi\)
							−0.455111	+	0.890435i	\(0.650400\pi\)
\(19\)	1.27320	−	2.20524i	0.292091	−	0.505917i	−0.682213	−	0.731154i	\(-0.738982\pi\)
							0.974304	+	0.225237i	\(0.0723155\pi\)
\(23\)	−4.05029	−	4.05029i	−0.844544	−	0.844544i	0.144902	−	0.989446i	\(-0.453713\pi\)
							−0.989446	+	0.144902i	\(0.953713\pi\)
\(29\)	−1.95299	+	1.95299i	−0.362660	+	0.362660i	−0.864791	−	0.502131i	\(-0.832549\pi\)
							0.502131	+	0.864791i	\(0.332549\pi\)
\(31\)	7.72184	−	4.45821i	1.38688	−	0.800718i	0.393921	−	0.919144i	\(-0.371118\pi\)
							0.992963	+	0.118427i	\(0.0377851\pi\)
\(37\)	−1.31245	−	2.27323i	−0.215765	−	0.373716i	0.737744	−	0.675081i	\(-0.235891\pi\)
							−0.953509	+	0.301365i	\(0.902558\pi\)
\(41\)	4.94092	−	4.94092i	0.771641	−	0.771641i	−0.206752	−	0.978393i	\(-0.566289\pi\)
							0.978393	+	0.206752i	\(0.0662893\pi\)
\(43\)	1.35062	−	5.04059i	0.205968	−	0.768683i	−0.783184	−	0.621790i	\(-0.786406\pi\)
							0.989152	−	0.146893i	\(-0.0469275\pi\)
\(47\)	6.54506	−	3.77879i	0.954696	−	0.551194i	0.0601593	−	0.998189i	\(-0.480839\pi\)
							0.894537	+	0.446995i	\(0.147506\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	471.2.l.b.50.16	✓	200
3.2	odd	2	inner	471.2.l.b.50.35	yes	200
157.22	odd	12	inner	471.2.l.b.179.35	yes	200
471.179	even	12	inner	471.2.l.b.179.16	yes	200

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
471.2.l.b.50.16	✓	200	1.1	even	1	trivial
471.2.l.b.50.35	yes	200	3.2	odd	2	inner
471.2.l.b.179.16	yes	200	471.179	even	12	inner
471.2.l.b.179.35	yes	200	157.22	odd	12	inner