Embedded newform 483.2.bf.a.61.9

• Label: 483.2.bf.a.61.9
• Level: $483$
• Weight: $2$
• Character: 483.61
• Analytic conductor: $3.857$
• Analytic rank: $0$
• Dimension: $640$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 483 = 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	483.bf (of order \(66\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.85677441763\)
Analytic rank:	\(0\)
Dimension:	\(640\)
Relative dimension:	\(32\) over \(\Q(\zeta_{66})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{66}]$

Embedding invariants

Embedding label			61.9
Character	\(\chi\)	\(=\)	483.61
Dual form			483.2.bf.a.388.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.34752 - 1.05970i) q^{2} +(0.814576 + 0.580057i) q^{3} +(0.221326 + 0.912318i) q^{4} +(-1.18796 + 0.228961i) q^{5} +(-0.482971 - 1.64485i) q^{6} +(0.252104 + 2.63371i) q^{7} +(-0.755737 + 1.65483i) q^{8} +(0.327068 + 0.945001i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q - 4 q^{2} + 36 q^{4} + 24 q^{8} - 32 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(323\)	\(346\)	\(442\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{17}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.34752	−	1.05970i	−0.952841	−	0.749323i	0.0152597	−	0.999884i	\(-0.495143\pi\)
							−0.968101	+	0.250561i	\(0.919385\pi\)
\(3\)	0.814576	+	0.580057i	0.470296	+	0.334896i
							\(5\)	−1.18796	+	0.228961i	−0.531272	+	0.102394i	−0.447831	−	0.894118i	\(-0.647803\pi\)
−0.0834414	+	0.996513i	\(0.526591\pi\)
\(7\)	0.252104	+	2.63371i	0.0952863	+	0.995450i
							\(11\)	−2.19945	−	2.79683i	−0.663160	−	0.843277i	0.331821	−	0.943342i	\(-0.392337\pi\)
−0.994981	+	0.100066i	\(0.968095\pi\)
\(13\)	−3.01057	−	4.68454i	−0.834982	−	1.29926i	−0.951991	−	0.306127i	\(-0.900967\pi\)
							0.117008	−	0.993131i	\(-0.462670\pi\)
\(17\)	−4.10833	−	3.91728i	−0.996416	−	0.950080i	0.00233765	−	0.999997i	\(-0.499256\pi\)
							−0.998753	+	0.0499168i	\(0.984104\pi\)
\(19\)	5.09250	−	4.85569i	1.16830	−	1.11397i	0.176748	−	0.984256i	\(-0.443442\pi\)
							0.991551	−	0.129715i	\(-0.0414061\pi\)
\(23\)	−1.33516	−	4.60623i	−0.278399	−	0.960465i
							\(29\)	1.64016	−	0.481594i	0.304570	−	0.0894298i	−0.125877	−	0.992046i	\(-0.540174\pi\)
0.430446	+	0.902616i	\(0.358356\pi\)
\(31\)	−0.0541600	−	0.567190i	−0.00972743	−	0.101870i	0.989304	−	0.145867i	\(-0.0465970\pi\)
							−0.999032	+	0.0439964i	\(0.985991\pi\)
\(37\)	0.0333173	−	0.0115312i	0.00547733	−	0.00189572i	−0.324328	−	0.945945i	\(-0.605138\pi\)
							0.329805	+	0.944049i	\(0.393017\pi\)
\(41\)	3.96851	−	3.43874i	0.619778	−	0.537040i	−0.287388	−	0.957814i	\(-0.592787\pi\)
							0.907166	+	0.420774i	\(0.138241\pi\)
\(43\)	1.02326	−	0.467307i	0.156046	−	0.0712636i	−0.335861	−	0.941911i	\(-0.609027\pi\)
							0.491907	+	0.870648i	\(0.336300\pi\)
\(47\)	−2.39693	−	1.38387i	−0.349627	−	0.201858i	0.314894	−	0.949127i	\(-0.398031\pi\)
							−0.664521	+	0.747269i	\(0.731364\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.bf.a.61.9	✓	640
7.3	odd	6	inner	483.2.bf.a.199.24	yes	640
23.20	odd	22	inner	483.2.bf.a.250.24	yes	640
161.66	even	66	inner	483.2.bf.a.388.9	yes	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.bf.a.61.9	✓	640	1.1	even	1	trivial
483.2.bf.a.199.24	yes	640	7.3	odd	6	inner
483.2.bf.a.250.24	yes	640	23.20	odd	22	inner
483.2.bf.a.388.9	yes	640	161.66	even	66	inner