Embedded newform 471.2.l.b.50.13

• Label: 471.2.l.b.50.13
• 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			50.13
Character	\(\chi\)	\(=\)	471.50
Dual form			471.2.l.b.179.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.17584 + 1.17584i) q^{2} +(-1.72956 + 0.0927798i) q^{3} -0.765185i q^{4} +(-0.933382 - 3.48343i) q^{5} +(1.92459 - 2.14278i) q^{6} +(-2.60917 + 2.60917i) q^{7} +(-1.45194 - 1.45194i) q^{8} +(2.98278 - 0.320937i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 200 q - 6 q^{3} - 10 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.17584	+	1.17584i	−0.831442	+	0.831442i	−0.987714	−	0.156272i	\(-0.950052\pi\)
							0.156272	+	0.987714i	\(0.450052\pi\)
\(3\)	−1.72956	+	0.0927798i	−0.998564	+	0.0535665i
							\(5\)	−0.933382	−	3.48343i	−0.417421	−	1.55784i	−0.779936	−	0.625859i	\(-0.784749\pi\)
0.362515	−	0.931978i	\(-0.381918\pi\)
\(7\)	−2.60917	+	2.60917i	−0.986174	+	0.986174i	−0.999906	−	0.0137313i	\(-0.995629\pi\)
							0.0137313	+	0.999906i	\(0.495629\pi\)
\(11\)	−3.28516	−	1.89669i	−0.990514	−	0.571873i	−0.0850860	−	0.996374i	\(-0.527117\pi\)
							−0.905428	+	0.424500i	\(0.860450\pi\)
\(13\)	0.125759	−	0.0726069i	0.0348792	−	0.0201375i	−0.482459	−	0.875919i	\(-0.660256\pi\)
							0.517338	+	0.855781i	\(0.326923\pi\)
\(17\)	2.82816	+	1.63284i	0.685929	+	0.396021i	0.802085	−	0.597210i	\(-0.203724\pi\)
							−0.116156	+	0.993231i	\(0.537057\pi\)
\(19\)	0.0429008	−	0.0743064i	0.00984212	−	0.0170471i	−0.861062	−	0.508499i	\(-0.830200\pi\)
							0.870905	+	0.491452i	\(0.163534\pi\)
\(23\)	5.72967	+	5.72967i	1.19472	+	1.19472i	0.975727	+	0.218991i	\(0.0702766\pi\)
							0.218991	+	0.975727i	\(0.429723\pi\)
\(29\)	1.49596	−	1.49596i	0.277793	−	0.277793i	−0.554435	−	0.832227i	\(-0.687065\pi\)
							0.832227	+	0.554435i	\(0.187065\pi\)
\(31\)	3.04409	−	1.75750i	0.546734	−	0.315657i	−0.201070	−	0.979577i	\(-0.564442\pi\)
							0.747804	+	0.663920i	\(0.231108\pi\)
\(37\)	2.91356	+	5.04643i	0.478986	+	0.829628i	0.999710	−	0.0240975i	\(-0.00767122\pi\)
							−0.520724	+	0.853725i	\(0.674338\pi\)
\(41\)	−0.710429	+	0.710429i	−0.110950	+	0.110950i	−0.760402	−	0.649452i	\(-0.774998\pi\)
							0.649452	+	0.760402i	\(0.274998\pi\)
\(43\)	0.0952042	−	0.355307i	0.0145185	−	0.0541838i	−0.958287	−	0.285809i	\(-0.907738\pi\)
							0.972805	+	0.231625i	\(0.0744043\pi\)
\(47\)	−11.1945	+	6.46314i	−1.63288	+	0.942746i	−0.649686	+	0.760202i	\(0.725100\pi\)
							−0.983198	+	0.182544i	\(0.941567\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	471.2.l.b.50.13	✓	200
3.2	odd	2	inner	471.2.l.b.50.38	yes	200
157.22	odd	12	inner	471.2.l.b.179.38	yes	200
471.179	even	12	inner	471.2.l.b.179.13	yes	200

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
471.2.l.b.50.13	✓	200	1.1	even	1	trivial
471.2.l.b.50.38	yes	200	3.2	odd	2	inner
471.2.l.b.179.13	yes	200	471.179	even	12	inner
471.2.l.b.179.38	yes	200	157.22	odd	12	inner