Embedded newform 471.2.l.b.50.8

• Label: 471.2.l.b.50.8
• 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			50.8
Character	\(\chi\)	\(=\)	471.50
Dual form			471.2.l.b.179.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.52089 + 1.52089i) q^{2} +(-0.258431 - 1.71266i) q^{3} -2.62620i q^{4} +(0.679058 + 2.53428i) q^{5} +(2.99781 + 2.21172i) q^{6} +(1.37133 - 1.37133i) q^{7} +(0.952382 + 0.952382i) q^{8} +(-2.86643 + 0.885212i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 200 q - 6 q^{3} - 10 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.52089	+	1.52089i	−1.07543	+	1.07543i	−0.0785176	+	0.996913i	\(0.525019\pi\)
							−0.996913	+	0.0785176i	\(0.974981\pi\)
\(3\)	−0.258431	−	1.71266i	−0.149205	−	0.988806i
							\(5\)	0.679058	+	2.53428i	0.303684	+	1.13336i	0.934072	+	0.357084i	\(0.116229\pi\)
−0.630388	+	0.776280i	\(0.717104\pi\)
\(7\)	1.37133	−	1.37133i	0.518313	−	0.518313i	−0.398748	−	0.917061i	\(-0.630555\pi\)
							0.917061	+	0.398748i	\(0.130555\pi\)
\(11\)	1.56312	+	0.902467i	0.471298	+	0.272104i	0.716783	−	0.697296i	\(-0.245614\pi\)
							−0.245485	+	0.969400i	\(0.578947\pi\)
\(13\)	−5.21185	+	3.00906i	−1.44551	+	0.834564i	−0.998209	−	0.0598210i	\(-0.980947\pi\)
							−0.447298	+	0.894385i	\(0.647614\pi\)
\(17\)	4.34307	+	2.50747i	1.05335	+	0.608151i	0.923585	−	0.383393i	\(-0.125245\pi\)
							0.129764	+	0.991545i	\(0.458578\pi\)
\(19\)	2.66745	−	4.62016i	0.611955	−	1.05994i	−0.378956	−	0.925415i	\(-0.623717\pi\)
							0.990911	−	0.134522i	\(-0.0429498\pi\)
\(23\)	2.72324	+	2.72324i	0.567836	+	0.567836i	0.931522	−	0.363686i	\(-0.118482\pi\)
							−0.363686	+	0.931522i	\(0.618482\pi\)
\(29\)	−1.76143	+	1.76143i	−0.327089	+	0.327089i	−0.851479	−	0.524389i	\(-0.824294\pi\)
							0.524389	+	0.851479i	\(0.324294\pi\)
\(31\)	−0.0204291	+	0.0117947i	−0.00366917	+	0.00211840i	−0.501833	−	0.864964i	\(-0.667341\pi\)
							0.498164	+	0.867083i	\(0.334008\pi\)
\(37\)	5.78725	+	10.0238i	0.951419	+	1.64791i	0.742358	+	0.670003i	\(0.233707\pi\)
							0.209060	+	0.977903i	\(0.432959\pi\)
\(41\)	−5.48492	+	5.48492i	−0.856600	+	0.856600i	−0.990936	−	0.134336i	\(-0.957110\pi\)
							0.134336	+	0.990936i	\(0.457110\pi\)
\(43\)	−0.795307	+	2.96813i	−0.121283	+	0.452635i	−0.999680	−	0.0252970i	\(-0.991947\pi\)
							0.878397	+	0.477932i	\(0.158614\pi\)
\(47\)	−0.753741	+	0.435172i	−0.109944	+	0.0634764i	−0.553964	−	0.832541i	\(-0.686885\pi\)
							0.444019	+	0.896017i	\(0.353552\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	471.2.l.b.50.8	✓	200
3.2	odd	2	inner	471.2.l.b.50.43	yes	200
157.22	odd	12	inner	471.2.l.b.179.43	yes	200
471.179	even	12	inner	471.2.l.b.179.8	yes	200

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
471.2.l.b.50.8	✓	200	1.1	even	1	trivial
471.2.l.b.50.43	yes	200	3.2	odd	2	inner
471.2.l.b.179.8	yes	200	471.179	even	12	inner
471.2.l.b.179.43	yes	200	157.22	odd	12	inner