Embedded newform 460.2.j.a.47.18

• Label: 460.2.j.a.47.18
• 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			47.18
Character	\(\chi\)	\(=\)	460.47
Dual form			460.2.j.a.323.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.865562 + 1.11839i) q^{2} +(-2.09555 - 2.09555i) q^{3} +(-0.501606 - 1.93608i) q^{4} +(-0.928013 + 2.03440i) q^{5} +(4.15748 - 0.529822i) q^{6} +(2.24593 - 2.24593i) q^{7} +(2.59947 + 1.11480i) q^{8} +5.78270i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 132 q - 12 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.865562	+	1.11839i	−0.612045	+	0.790823i
							\(3\)	−2.09555	−	2.09555i	−1.20987	−	1.20987i	−0.971069	−	0.238800i	\(-0.923246\pi\)
−0.238800	−	0.971069i	\(-0.576754\pi\)
\(5\)	−0.928013	+	2.03440i	−0.415020	+	0.909812i
							\(7\)	2.24593	−	2.24593i	0.848880	−	0.848880i	−0.141114	−	0.989993i	\(-0.545068\pi\)
0.989993	+	0.141114i	\(0.0450683\pi\)
\(11\)			0.112800i			0.0340106i	0.999855	+	0.0170053i	\(0.00541322\pi\)
							−0.999855	+	0.0170053i	\(0.994587\pi\)
\(13\)	3.49630	−	3.49630i	0.969699	−	0.969699i	−0.0298551	−	0.999554i	\(-0.509505\pi\)
							0.999554	+	0.0298551i	\(0.00950458\pi\)
\(17\)	0.986494	+	0.986494i	0.239260	+	0.239260i	0.816544	−	0.577284i	\(-0.195887\pi\)
							−0.577284	+	0.816544i	\(0.695887\pi\)
\(19\)	−6.57035			−1.50734			−0.753671	−	0.657251i	\(-0.771719\pi\)
							−0.753671	+	0.657251i	\(0.771719\pi\)
\(23\)	−0.707107	−	0.707107i	−0.147442	−	0.147442i
							\(29\)		−	6.73812i		−	1.25124i	−0.780129	−	0.625619i	\(-0.784847\pi\)
0.780129	−	0.625619i	\(-0.215153\pi\)
\(31\)			0.104390i			0.0187490i	0.999956	+	0.00937452i	\(0.00298405\pi\)
							−0.999956	+	0.00937452i	\(0.997016\pi\)
\(37\)	−8.41485	−	8.41485i	−1.38339	−	1.38339i	−0.838516	−	0.544877i	\(-0.816576\pi\)
							−0.544877	−	0.838516i	\(-0.683424\pi\)
\(41\)	−4.68563			−0.731772			−0.365886	−	0.930660i	\(-0.619234\pi\)
							−0.365886	+	0.930660i	\(0.619234\pi\)
\(43\)	−0.312518	−	0.312518i	−0.0476586	−	0.0476586i	0.682876	−	0.730534i	\(-0.260729\pi\)
							−0.730534	+	0.682876i	\(0.760729\pi\)
\(47\)	4.32182	−	4.32182i	0.630402	−	0.630402i	−0.317767	−	0.948169i	\(-0.602933\pi\)
							0.948169	+	0.317767i	\(0.102933\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	460.2.j.a.47.18	✓	132
4.3	odd	2	inner	460.2.j.a.47.53	yes	132
5.3	odd	4	inner	460.2.j.a.323.53	yes	132
20.3	even	4	inner	460.2.j.a.323.18	yes	132

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
460.2.j.a.47.18	✓	132	1.1	even	1	trivial
460.2.j.a.47.53	yes	132	4.3	odd	2	inner
460.2.j.a.323.18	yes	132	20.3	even	4	inner
460.2.j.a.323.53	yes	132	5.3	odd	4	inner