Embedded newform 483.2.n.a.47.2

• Label: 483.2.n.a.47.2
• Level: $483$
• Weight: $2$
• Character: 483.47
• Analytic conductor: $3.857$
• Analytic rank: $0$
• Dimension: $116$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 483 = 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	483.n (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.85677441763\)
Analytic rank:	\(0\)
Dimension:	\(116\)
Relative dimension:	\(58\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			47.2
Character	\(\chi\)	\(=\)	483.47
Dual form			483.2.n.a.185.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.33544 - 1.34837i) q^{2} +(-1.04040 + 1.38476i) q^{3} +(2.63619 + 4.56602i) q^{4} +(-0.0332776 + 0.0576385i) q^{5} +(4.29696 - 1.83120i) q^{6} +(-1.58536 + 2.11817i) q^{7} -8.82477i q^{8} +(-0.835146 - 2.88141i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 116 q + 56 q^{4} - 10 q^{7} - 4 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/483\mathbb{Z}\right)^\times\).

\(n\)	\(323\)	\(346\)	\(442\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{5}{6}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.33544	−	1.34837i	−1.65141	−	0.953440i	−0.976494	−	0.215543i	\(-0.930848\pi\)
							−0.674913	−	0.737897i	\(-0.735819\pi\)
\(3\)	−1.04040	+	1.38476i	−0.600674	+	0.799494i
							\(5\)	−0.0332776	+	0.0576385i	−0.0148822	+	0.0257767i	−0.873371	−	0.487056i	\(-0.838071\pi\)
0.858488	+	0.512833i	\(0.171404\pi\)
\(7\)	−1.58536	+	2.11817i	−0.599211	+	0.800591i
							\(11\)	2.92294	−	1.68756i	0.881298	−	0.508818i	0.0102122	−	0.999948i	\(-0.496749\pi\)
0.871086	+	0.491130i	\(0.163416\pi\)
\(13\)		−	5.15668i		−	1.43021i	−0.699019	−	0.715103i	\(-0.746380\pi\)
							0.699019	−	0.715103i	\(-0.253620\pi\)
\(17\)	−1.02080	−	1.76808i	−0.247581	−	0.428823i	0.715273	−	0.698845i	\(-0.246302\pi\)
							−0.962854	+	0.270022i	\(0.912969\pi\)
\(19\)	4.31598	+	2.49183i	0.990154	+	0.571666i	0.905320	−	0.424729i	\(-0.139631\pi\)
							0.0848336	+	0.996395i	\(0.472964\pi\)
\(23\)	−0.866025	−	0.500000i	−0.180579	−	0.104257i
							\(29\)			7.45496i			1.38435i	0.721729	+	0.692175i	\(0.243348\pi\)
−0.721729	+	0.692175i	\(0.756652\pi\)
\(31\)	1.20831	−	0.697619i	0.217019	−	0.125296i	−0.387550	−	0.921849i	\(-0.626679\pi\)
							0.604569	+	0.796553i	\(0.293345\pi\)
\(37\)	−0.946344	+	1.63912i	−0.155578	+	0.269469i	−0.933269	−	0.359177i	\(-0.883057\pi\)
							0.777691	+	0.628646i	\(0.216391\pi\)
\(41\)	1.12245			0.175298			0.0876489	−	0.996151i	\(-0.472065\pi\)
							0.0876489	+	0.996151i	\(0.472065\pi\)
\(43\)	10.7549			1.64010			0.820051	−	0.572291i	\(-0.193945\pi\)
							0.820051	+	0.572291i	\(0.193945\pi\)
\(47\)	1.33865	−	2.31861i	0.195262	−	0.338203i	−0.751725	−	0.659477i	\(-0.770778\pi\)
							0.946986	+	0.321274i	\(0.104111\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.n.a.47.2	✓	116
3.2	odd	2	inner	483.2.n.a.47.57	yes	116
7.3	odd	6	inner	483.2.n.a.185.57	yes	116
21.17	even	6	inner	483.2.n.a.185.2	yes	116

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.n.a.47.2	✓	116	1.1	even	1	trivial
483.2.n.a.47.57	yes	116	3.2	odd	2	inner
483.2.n.a.185.2	yes	116	21.17	even	6	inner
483.2.n.a.185.57	yes	116	7.3	odd	6	inner