Embedded newform 471.2.q.b.19.10

• Label: 471.2.q.b.19.10
• 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			19.10
Character	\(\chi\)	\(=\)	471.19
Dual form			471.2.q.b.124.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.12662 + 0.277686i) q^{2} +(-0.845190 - 0.534466i) q^{3} +(-0.578755 - 0.303754i) q^{4} +(2.59048 - 1.94662i) q^{5} +(-0.803792 - 0.836836i) q^{6} +(-2.02054 + 1.06046i) q^{7} +(-2.30473 - 2.04182i) q^{8} +(0.428693 + 0.903450i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 2 q^{2} + 14 q^{3} - 26 q^{4} + 2 q^{5} + q^{6} - 4 q^{7} + 14 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.12662	+	0.277686i	0.796639	+	0.196354i	0.616570	−	0.787300i	\(-0.288522\pi\)
							0.180069	+	0.983654i	\(0.442368\pi\)
\(3\)	−0.845190	−	0.534466i	−0.487971	−	0.308574i
							\(5\)	2.59048	−	1.94662i	1.15850	−	0.870556i	0.165215	−	0.986258i	\(-0.447168\pi\)
0.993284	+	0.115702i	\(0.0369117\pi\)
\(7\)	−2.02054	+	1.06046i	−0.763693	+	0.400817i	−0.801137	−	0.598482i	\(-0.795771\pi\)
							0.0374433	+	0.999299i	\(0.488079\pi\)
\(11\)	−0.677222	−	0.705063i	−0.204190	−	0.212584i	0.611232	−	0.791452i	\(-0.290674\pi\)
							−0.815422	+	0.578867i	\(0.803495\pi\)
\(13\)	3.43943	−	5.95727i	0.953926	−	1.65225i	0.217119	−	0.976145i	\(-0.430334\pi\)
							0.736807	−	0.676103i	\(-0.236333\pi\)
\(17\)	−2.37287	+	1.78310i	−0.575506	+	0.432465i	−0.848230	−	0.529628i	\(-0.822332\pi\)
							0.272724	+	0.962092i	\(0.412075\pi\)
\(19\)	0.213666	−	1.04661i	0.0490184	−	0.240108i	−0.947899	−	0.318572i	\(-0.896797\pi\)
							0.996917	+	0.0784645i	\(0.0250017\pi\)
\(23\)	−3.48625	−	5.05070i	−0.726933	−	1.05314i	−0.995941	−	0.0900063i	\(-0.971311\pi\)
							0.269008	−	0.963138i	\(-0.413304\pi\)
\(29\)	0.263214	−	2.16776i	0.0488776	−	0.402543i	−0.947652	−	0.319305i	\(-0.896551\pi\)
							0.996530	−	0.0832383i	\(-0.0265263\pi\)
\(31\)	0.554986	+	0.679735i	0.0996785	+	0.122084i	0.822035	−	0.569437i	\(-0.192839\pi\)
							−0.722357	+	0.691521i	\(0.756941\pi\)
\(37\)	5.91973	+	4.44839i	0.973197	+	0.731310i	0.963241	−	0.268640i	\(-0.0865743\pi\)
							0.00995645	+	0.999950i	\(0.496831\pi\)
\(41\)	1.11260	−	1.61188i	0.173759	−	0.251733i	−0.726423	−	0.687248i	\(-0.758818\pi\)
							0.900181	+	0.435515i	\(0.143434\pi\)
\(43\)	−1.27702	+	0.807538i	−0.194744	+	0.123148i	−0.628279	−	0.777988i	\(-0.716240\pi\)
							0.433535	+	0.901137i	\(0.357266\pi\)
\(47\)	9.90559	+	4.22038i	1.44488	+	0.615605i	0.965123	−	0.261797i	\(-0.0843152\pi\)
							0.479755	+	0.877402i	\(0.340725\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	471.2.q.b.19.10	✓	336
157.124	even	39	inner	471.2.q.b.124.10	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
471.2.q.b.19.10	✓	336	1.1	even	1	trivial
471.2.q.b.124.10	yes	336	157.124	even	39	inner