Embedded newform 460.2.j.a.47.1

• Label: 460.2.j.a.47.1
• 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			47.1
Character	\(\chi\)	\(=\)	460.47
Dual form			460.2.j.a.323.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.41322 + 0.0530604i) q^{2} +(1.60974 + 1.60974i) q^{3} +(1.99437 - 0.149972i) q^{4} +(-1.92782 - 1.13292i) q^{5} +(-2.36032 - 2.18950i) q^{6} +(1.60901 - 1.60901i) q^{7} +(-2.81052 + 0.317765i) q^{8} +2.18251i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 132 q - 12 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.41322	+	0.0530604i	−0.999296	+	0.0375194i
							\(3\)	1.60974	+	1.60974i	0.929382	+	0.929382i	0.997666	−	0.0682839i	\(-0.0217524\pi\)
−0.0682839	+	0.997666i	\(0.521752\pi\)
\(5\)	−1.92782	−	1.13292i	−0.862148	−	0.506657i
							\(7\)	1.60901	−	1.60901i	0.608150	−	0.608150i	−0.334312	−	0.942462i	\(-0.608504\pi\)
0.942462	+	0.334312i	\(0.108504\pi\)
\(11\)			4.49586i			1.35555i	0.735268	+	0.677776i	\(0.237056\pi\)
							−0.735268	+	0.677776i	\(0.762944\pi\)
\(13\)	0.644962	−	0.644962i	0.178880	−	0.178880i	−0.611987	−	0.790868i	\(-0.709630\pi\)
							0.790868	+	0.611987i	\(0.209630\pi\)
\(17\)	4.52415	+	4.52415i	1.09727	+	1.09727i	0.994729	+	0.102537i	\(0.0326961\pi\)
							0.102537	+	0.994729i	\(0.467304\pi\)
\(19\)	6.06042			1.39036			0.695178	−	0.718838i	\(-0.255326\pi\)
							0.695178	+	0.718838i	\(0.255326\pi\)
\(23\)	−0.707107	−	0.707107i	−0.147442	−	0.147442i
							\(29\)			1.43565i			0.266594i	0.991076	+	0.133297i	\(0.0425564\pi\)
−0.991076	+	0.133297i	\(0.957444\pi\)
\(31\)			6.45921i			1.16011i	0.814578	+	0.580054i	\(0.196969\pi\)
							−0.814578	+	0.580054i	\(0.803031\pi\)
\(37\)	−5.84858	−	5.84858i	−0.961501	−	0.961501i	0.0377846	−	0.999286i	\(-0.487970\pi\)
							−0.999286	+	0.0377846i	\(0.987970\pi\)
\(41\)	5.67766			0.886702			0.443351	−	0.896348i	\(-0.353790\pi\)
							0.443351	+	0.896348i	\(0.353790\pi\)
\(43\)	−5.05142	−	5.05142i	−0.770334	−	0.770334i	0.207830	−	0.978165i	\(-0.433360\pi\)
							−0.978165	+	0.207830i	\(0.933360\pi\)
\(47\)	6.75592	−	6.75592i	0.985452	−	0.985452i	−0.0144441	−	0.999896i	\(-0.504598\pi\)
							0.999896	+	0.0144441i	\(0.00459785\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	460.2.j.a.47.1	✓	132
4.3	odd	2	inner	460.2.j.a.47.33	yes	132
5.3	odd	4	inner	460.2.j.a.323.33	yes	132
20.3	even	4	inner	460.2.j.a.323.1	yes	132

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
460.2.j.a.47.1	✓	132	1.1	even	1	trivial
460.2.j.a.47.33	yes	132	4.3	odd	2	inner
460.2.j.a.323.1	yes	132	20.3	even	4	inner
460.2.j.a.323.33	yes	132	5.3	odd	4	inner