Embedded newform 460.2.j.a.47.19

• Label: 460.2.j.a.47.19
• Level: $460$
• Weight: $2$
• Character: 460.47
• Analytic conductor: $3.673$
• Analytic rank: $0$
• Dimension: $132$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 460 = 2^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	460.j (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.67311849298\)
Analytic rank:	\(0\)
Dimension:	\(132\)
Relative dimension:	\(66\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			47.19
Character	\(\chi\)	\(=\)	460.47
Dual form			460.2.j.a.323.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.816781 - 1.15450i) q^{2} +(2.22730 + 2.22730i) q^{3} +(-0.665738 + 1.88595i) q^{4} +(-0.469590 + 2.18620i) q^{5} +(0.752200 - 4.39063i) q^{6} +(3.32421 - 3.32421i) q^{7} +(2.72109 - 0.771810i) q^{8} +6.92171i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 132 q - 12 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/460\mathbb{Z}\right)^\times\).

\(n\)	\(231\)	\(277\)	\(281\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{4}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.816781	−	1.15450i	−0.577551	−	0.816354i
							\(3\)	2.22730	+	2.22730i	1.28593	+	1.28593i	0.937236	+	0.348695i	\(0.113375\pi\)
0.348695	+	0.937236i	\(0.386625\pi\)
\(5\)	−0.469590	+	2.18620i	−0.210007	+	0.977700i
							\(7\)	3.32421	−	3.32421i	1.25643	−	1.25643i	0.303651	−	0.952783i	\(-0.401794\pi\)
0.952783	−	0.303651i	\(-0.0982057\pi\)
\(11\)			3.37738i			1.01832i	0.860673	+	0.509159i	\(0.170043\pi\)
							−0.860673	+	0.509159i	\(0.829957\pi\)
\(13\)	2.45773	−	2.45773i	0.681653	−	0.681653i	−0.278720	−	0.960372i	\(-0.589910\pi\)
							0.960372	+	0.278720i	\(0.0899101\pi\)
\(17\)	−2.25642	−	2.25642i	−0.547263	−	0.547263i	0.378385	−	0.925648i	\(-0.376480\pi\)
							−0.925648	+	0.378385i	\(0.876480\pi\)
\(19\)	−2.76777			−0.634970			−0.317485	−	0.948263i	\(-0.602838\pi\)
							−0.317485	+	0.948263i	\(0.602838\pi\)
\(23\)	−0.707107	−	0.707107i	−0.147442	−	0.147442i
							\(29\)		−	0.216946i		−	0.0402859i	−0.999797	−	0.0201430i	\(-0.993588\pi\)
0.999797	−	0.0201430i	\(-0.00641214\pi\)
\(31\)			0.718887i			0.129116i	0.997914	+	0.0645579i	\(0.0205637\pi\)
							−0.997914	+	0.0645579i	\(0.979436\pi\)
\(37\)	3.09799	+	3.09799i	0.509307	+	0.509307i	0.914314	−	0.405007i	\(-0.132731\pi\)
							−0.405007	+	0.914314i	\(0.632731\pi\)
\(41\)	5.19371			0.811121			0.405561	−	0.914068i	\(-0.367076\pi\)
							0.405561	+	0.914068i	\(0.367076\pi\)
\(43\)	−0.833042	−	0.833042i	−0.127038	−	0.127038i	0.640729	−	0.767767i	\(-0.278632\pi\)
							−0.767767	+	0.640729i	\(0.778632\pi\)
\(47\)	−5.71302	+	5.71302i	−0.833330	+	0.833330i	−0.987971	−	0.154641i	\(-0.950578\pi\)
							0.154641	+	0.987971i	\(0.450578\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	460.2.j.a.47.19	yes	132
4.3	odd	2	inner	460.2.j.a.47.15	✓	132
5.3	odd	4	inner	460.2.j.a.323.15	yes	132
20.3	even	4	inner	460.2.j.a.323.19	yes	132

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
460.2.j.a.47.15	✓	132	4.3	odd	2	inner
460.2.j.a.47.19	yes	132	1.1	even	1	trivial
460.2.j.a.323.15	yes	132	5.3	odd	4	inner
460.2.j.a.323.19	yes	132	20.3	even	4	inner