Embedded newform 471.2.l.b.50.19

• Label: 471.2.l.b.50.19
• 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			50.19
Character	\(\chi\)	\(=\)	471.50
Dual form			471.2.l.b.179.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.640983 + 0.640983i) q^{2} +(0.167689 + 1.72391i) q^{3} +1.17828i q^{4} +(0.280265 + 1.04596i) q^{5} +(-1.21249 - 0.997514i) q^{6} +(0.268331 - 0.268331i) q^{7} +(-2.03722 - 2.03722i) q^{8} +(-2.94376 + 0.578164i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 200 q - 6 q^{3} - 10 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.640983	+	0.640983i	−0.453244	+	0.453244i	−0.896430	−	0.443186i	\(-0.853848\pi\)
							0.443186	+	0.896430i	\(0.353848\pi\)
\(3\)	0.167689	+	1.72391i	0.0968155	+	0.995302i
							\(5\)	0.280265	+	1.04596i	0.125338	+	0.467769i	0.999852	−	0.0172328i	\(-0.00548565\pi\)
−0.874513	+	0.485002i	\(0.838819\pi\)
\(7\)	0.268331	−	0.268331i	0.101420	−	0.101420i	−0.654576	−	0.755996i	\(-0.727153\pi\)
							0.755996	+	0.654576i	\(0.227153\pi\)
\(11\)	−5.58070	−	3.22202i	−1.68264	−	0.971475i	−0.959893	−	0.280368i	\(-0.909543\pi\)
							−0.722752	−	0.691107i	\(-0.757123\pi\)
\(13\)	−2.54794	+	1.47106i	−0.706672	+	0.407997i	−0.809828	−	0.586668i	\(-0.800439\pi\)
							0.103155	+	0.994665i	\(0.467106\pi\)
\(17\)	5.06477	+	2.92414i	1.22839	+	0.709209i	0.966692	−	0.255942i	\(-0.0823855\pi\)
							0.261694	+	0.965151i	\(0.415719\pi\)
\(19\)	0.166844	−	0.288983i	0.0382767	−	0.0662972i	−0.846252	−	0.532782i	\(-0.821147\pi\)
							0.884529	+	0.466485i	\(0.154480\pi\)
\(23\)	3.29836	+	3.29836i	0.687756	+	0.687756i	0.961736	−	0.273980i	\(-0.0883401\pi\)
							−0.273980	+	0.961736i	\(0.588340\pi\)
\(29\)	−5.10758	+	5.10758i	−0.948454	+	0.948454i	−0.998735	−	0.0502814i	\(-0.983988\pi\)
							0.0502814	+	0.998735i	\(0.483988\pi\)
\(31\)	−5.28974	+	3.05404i	−0.950066	+	0.548521i	−0.893101	−	0.449855i	\(-0.851476\pi\)
							−0.0569646	+	0.998376i	\(0.518142\pi\)
\(37\)	−2.70946	−	4.69292i	−0.445432	−	0.771512i	0.552650	−	0.833414i	\(-0.313617\pi\)
							−0.998082	+	0.0619019i	\(0.980283\pi\)
\(41\)	1.39669	−	1.39669i	0.218126	−	0.218126i	−0.589582	−	0.807708i	\(-0.700708\pi\)
							0.807708	+	0.589582i	\(0.200708\pi\)
\(43\)	−3.26274	+	12.1767i	−0.497563	+	1.85693i	0.0176138	+	0.999845i	\(0.494393\pi\)
							−0.515176	+	0.857084i	\(0.672274\pi\)
\(47\)	−0.929843	+	0.536845i	−0.135632	+	0.0783069i	−0.566280	−	0.824213i	\(-0.691618\pi\)
							0.430649	+	0.902520i	\(0.358285\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	471.2.l.b.50.19	✓	200
3.2	odd	2	inner	471.2.l.b.50.32	yes	200
157.22	odd	12	inner	471.2.l.b.179.32	yes	200
471.179	even	12	inner	471.2.l.b.179.19	yes	200

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
471.2.l.b.50.19	✓	200	1.1	even	1	trivial
471.2.l.b.50.32	yes	200	3.2	odd	2	inner
471.2.l.b.179.19	yes	200	471.179	even	12	inner
471.2.l.b.179.32	yes	200	157.22	odd	12	inner