Embedded newform 471.2.l.b.50.9

• Label: 471.2.l.b.50.9
• 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			50.9
Character	\(\chi\)	\(=\)	471.50
Dual form			471.2.l.b.179.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.47058 + 1.47058i) q^{2} +(1.73106 + 0.0586068i) q^{3} -2.32524i q^{4} +(-0.337319 - 1.25889i) q^{5} +(-2.63185 + 2.45948i) q^{6} +(0.411952 - 0.411952i) q^{7} +(0.478287 + 0.478287i) q^{8} +(2.99313 + 0.202904i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 200 q - 6 q^{3} - 10 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.47058	+	1.47058i	−1.03986	+	1.03986i	−0.0406882	+	0.999172i	\(0.512955\pi\)
							−0.999172	+	0.0406882i	\(0.987045\pi\)
\(3\)	1.73106	+	0.0586068i	0.999427	+	0.0338367i
							\(5\)	−0.337319	−	1.25889i	−0.150853	−	0.562993i	−0.999425	−	0.0339118i	\(-0.989203\pi\)
0.848571	−	0.529081i	\(-0.177463\pi\)
\(7\)	0.411952	−	0.411952i	0.155703	−	0.155703i	−0.624956	−	0.780660i	\(-0.714883\pi\)
							0.780660	+	0.624956i	\(0.214883\pi\)
\(11\)	−0.760356	−	0.438992i	−0.229256	−	0.132361i	0.380973	−	0.924586i	\(-0.375589\pi\)
							−0.610229	+	0.792225i	\(0.708922\pi\)
\(13\)	0.391301	−	0.225918i	0.108527	−	0.0626584i	−0.444754	−	0.895653i	\(-0.646709\pi\)
							0.553281	+	0.832995i	\(0.313375\pi\)
\(17\)	2.13230	+	1.23108i	0.517158	+	0.298581i	0.735771	−	0.677230i	\(-0.236820\pi\)
							−0.218613	+	0.975812i	\(0.570153\pi\)
\(19\)	0.720549	−	1.24803i	0.165305	−	0.286317i	−0.771458	−	0.636280i	\(-0.780472\pi\)
							0.936764	+	0.349963i	\(0.113806\pi\)
\(23\)	4.57832	+	4.57832i	0.954647	+	0.954647i	0.999015	−	0.0443686i	\(-0.0141276\pi\)
							−0.0443686	+	0.999015i	\(0.514128\pi\)
\(29\)	4.99378	−	4.99378i	0.927322	−	0.927322i	−0.0702105	−	0.997532i	\(-0.522367\pi\)
							0.997532	+	0.0702105i	\(0.0223671\pi\)
\(31\)	2.80888	−	1.62171i	0.504490	−	0.291268i	−0.226076	−	0.974110i	\(-0.572590\pi\)
							0.730566	+	0.682842i	\(0.239256\pi\)
\(37\)	−3.88259	−	6.72484i	−0.638294	−	1.10556i	−0.985807	−	0.167882i	\(-0.946307\pi\)
							0.347513	−	0.937675i	\(-0.387026\pi\)
\(41\)	−2.86027	+	2.86027i	−0.446700	+	0.446700i	−0.894256	−	0.447556i	\(-0.852294\pi\)
							0.447556	+	0.894256i	\(0.352294\pi\)
\(43\)	0.228867	−	0.854145i	0.0349019	−	0.130256i	−0.946277	−	0.323358i	\(-0.895188\pi\)
							0.981179	+	0.193102i	\(0.0618549\pi\)
\(47\)	2.91758	−	1.68447i	0.425573	−	0.245705i	−0.271886	−	0.962330i	\(-0.587647\pi\)
							0.697459	+	0.716625i	\(0.254314\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	471.2.l.b.50.9	✓	200
3.2	odd	2	inner	471.2.l.b.50.42	yes	200
157.22	odd	12	inner	471.2.l.b.179.42	yes	200
471.179	even	12	inner	471.2.l.b.179.9	yes	200

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
471.2.l.b.50.9	✓	200	1.1	even	1	trivial
471.2.l.b.50.42	yes	200	3.2	odd	2	inner
471.2.l.b.179.9	yes	200	471.179	even	12	inner
471.2.l.b.179.42	yes	200	157.22	odd	12	inner