Embedded newform 460.2.j.a.47.11

• Label: 460.2.j.a.47.11
• 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			47.11
Character	\(\chi\)	\(=\)	460.47
Dual form			460.2.j.a.323.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.29378 + 0.571084i) q^{2} +(-0.0565025 - 0.0565025i) q^{3} +(1.34773 - 1.47771i) q^{4} +(2.13204 - 0.674090i) q^{5} +(0.105369 + 0.0408341i) q^{6} +(2.63515 - 2.63515i) q^{7} +(-0.899763 + 2.68150i) q^{8} -2.99361i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 132 q - 12 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.29378	+	0.571084i	−0.914840	+	0.403817i
							\(3\)	−0.0565025	−	0.0565025i	−0.0326217	−	0.0326217i	0.690608	−	0.723229i	\(-0.257343\pi\)
−0.723229	+	0.690608i	\(0.757343\pi\)
\(5\)	2.13204	−	0.674090i	0.953478	−	0.301462i
							\(7\)	2.63515	−	2.63515i	0.995995	−	0.995995i	−0.00399738	−	0.999992i	\(-0.501272\pi\)
0.999992	+	0.00399738i	\(0.00127241\pi\)
\(11\)			4.75975i			1.43512i	0.696498	+	0.717559i	\(0.254741\pi\)
							−0.696498	+	0.717559i	\(0.745259\pi\)
\(13\)	2.34974	−	2.34974i	0.651700	−	0.651700i	−0.301702	−	0.953402i	\(-0.597555\pi\)
							0.953402	+	0.301702i	\(0.0975549\pi\)
\(17\)	−5.32758	−	5.32758i	−1.29213	−	1.29213i	−0.933468	−	0.358661i	\(-0.883233\pi\)
							−0.358661	−	0.933468i	\(-0.616767\pi\)
\(19\)	−4.43914			−1.01841			−0.509204	−	0.860646i	\(-0.670060\pi\)
							−0.509204	+	0.860646i	\(0.670060\pi\)
\(23\)	−0.707107	−	0.707107i	−0.147442	−	0.147442i
							\(29\)			6.76486i			1.25620i	0.778131	+	0.628102i	\(0.216168\pi\)
−0.778131	+	0.628102i	\(0.783832\pi\)
\(31\)			1.89037i			0.339521i	0.985485	+	0.169760i	\(0.0542993\pi\)
							−0.985485	+	0.169760i	\(0.945701\pi\)
\(37\)	3.93394	+	3.93394i	0.646736	+	0.646736i	0.952203	−	0.305467i	\(-0.0988124\pi\)
							−0.305467	+	0.952203i	\(0.598812\pi\)
\(41\)	2.74342			0.428450			0.214225	−	0.976784i	\(-0.431277\pi\)
							0.214225	+	0.976784i	\(0.431277\pi\)
\(43\)	−3.40057	−	3.40057i	−0.518582	−	0.518582i	0.398560	−	0.917142i	\(-0.369510\pi\)
							−0.917142	+	0.398560i	\(0.869510\pi\)
\(47\)	9.32723	−	9.32723i	1.36052	−	1.36052i	0.487260	−	0.873257i	\(-0.337996\pi\)
							0.873257	−	0.487260i	\(-0.162004\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	460.2.j.a.47.11	✓	132
4.3	odd	2	inner	460.2.j.a.47.42	yes	132
5.3	odd	4	inner	460.2.j.a.323.42	yes	132
20.3	even	4	inner	460.2.j.a.323.11	yes	132

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
460.2.j.a.47.11	✓	132	1.1	even	1	trivial
460.2.j.a.47.42	yes	132	4.3	odd	2	inner
460.2.j.a.323.11	yes	132	20.3	even	4	inner
460.2.j.a.323.42	yes	132	5.3	odd	4	inner