Embedded newform 471.2.q.b.19.9

• Label: 471.2.q.b.19.9
• Level: $471$
• Weight: $2$
• Character: 471.19
• Analytic conductor: $3.761$
• Analytic rank: $0$
• Dimension: $336$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 471 = 3 \cdot 157 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	471.q (of order \(39\), degree \(24\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.76095393520\)
Analytic rank:	\(0\)
Dimension:	\(336\)
Relative dimension:	\(14\) over \(\Q(\zeta_{39})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{39}]$

Embedding invariants

Embedding label			19.9
Character	\(\chi\)	\(=\)	471.19
Dual form			471.2.q.b.124.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.309649 + 0.0763217i) q^{2} +(-0.845190 - 0.534466i) q^{3} +(-1.68085 - 0.882180i) q^{4} +(-2.49078 + 1.87170i) q^{5} +(-0.220921 - 0.230003i) q^{6} +(1.37070 - 0.719396i) q^{7} +(-0.930571 - 0.824414i) q^{8} +(0.428693 + 0.903450i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 2 q^{2} + 14 q^{3} - 26 q^{4} + 2 q^{5} + q^{6} - 4 q^{7} + 14 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(158\)	\(319\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{31}{39}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.309649	+	0.0763217i	0.218955	+	0.0539676i	0.347267	−	0.937766i	\(-0.387110\pi\)
							−0.128312	+	0.991734i	\(0.540956\pi\)
\(3\)	−0.845190	−	0.534466i	−0.487971	−	0.308574i
							\(5\)	−2.49078	+	1.87170i	−1.11391	+	0.837051i	−0.987918	−	0.154980i	\(-0.950469\pi\)
−0.125995	+	0.992031i	\(0.540212\pi\)
\(7\)	1.37070	−	0.719396i	0.518074	−	0.271906i	−0.185355	−	0.982672i	\(-0.559344\pi\)
							0.703429	+	0.710765i	\(0.251651\pi\)
\(11\)	3.51190	+	3.65627i	1.05888	+	1.10241i	0.994527	+	0.104481i	\(0.0333180\pi\)
							0.0643503	+	0.997927i	\(0.479503\pi\)
\(13\)	0.935096	−	1.61963i	0.259349	−	0.449206i	−0.706719	−	0.707495i	\(-0.749825\pi\)
							0.966068	+	0.258289i	\(0.0831587\pi\)
\(17\)	3.45010	−	2.59258i	0.836773	−	0.628794i	−0.0935179	−	0.995618i	\(-0.529811\pi\)
							0.930290	+	0.366824i	\(0.119555\pi\)
\(19\)	1.33802	−	6.55405i	0.306963	−	1.50360i	−0.474838	−	0.880073i	\(-0.657493\pi\)
							0.781800	−	0.623529i	\(-0.214302\pi\)
\(23\)	2.83721	+	4.11041i	0.591599	+	0.857079i	0.998406	−	0.0564349i	\(-0.0179733\pi\)
							−0.406807	+	0.913514i	\(0.633358\pi\)
\(29\)	0.408353	−	3.36309i	0.0758293	−	0.624510i	−0.904406	−	0.426674i	\(-0.859685\pi\)
							0.980235	−	0.197837i	\(-0.0633916\pi\)
\(31\)	3.19829	+	3.91719i	0.574430	+	0.703549i	0.977336	−	0.211692i	\(-0.0678975\pi\)
							−0.402906	+	0.915241i	\(0.632000\pi\)
\(37\)	5.31267	+	3.99221i	0.873397	+	0.656315i	0.939892	−	0.341471i	\(-0.110925\pi\)
							−0.0664950	+	0.997787i	\(0.521182\pi\)
\(41\)	−3.58883	+	5.19932i	−0.560482	+	0.811998i	−0.996109	−	0.0881294i	\(-0.971911\pi\)
							0.435628	+	0.900127i	\(0.356527\pi\)
\(43\)	−4.40540	+	2.78580i	−0.671817	+	0.424831i	−0.826350	−	0.563156i	\(-0.809587\pi\)
							0.154534	+	0.987988i	\(0.450612\pi\)
\(47\)	3.95883	+	1.68670i	0.577454	+	0.246030i	0.660890	−	0.750483i	\(-0.270179\pi\)
							−0.0834355	+	0.996513i	\(0.526589\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	471.2.q.b.19.9	✓	336
157.124	even	39	inner	471.2.q.b.124.9	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
471.2.q.b.19.9	✓	336	1.1	even	1	trivial
471.2.q.b.124.9	yes	336	157.124	even	39	inner