Embedded newform 483.2.bf.a.10.6

• Label: 483.2.bf.a.10.6
• Level: $483$
• Weight: $2$
• Character: 483.10
• 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			10.6
Character	\(\chi\)	\(=\)	483.10
Dual form			483.2.bf.a.145.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.643746 + 1.85998i) q^{2} +(-0.458227 - 0.888835i) q^{3} +(-1.47302 - 1.15839i) q^{4} +(2.63722 - 0.251824i) q^{5} +(1.94820 - 0.280109i) q^{6} +(-0.893381 - 2.49036i) q^{7} +(-0.208726 + 0.134140i) q^{8} +(-0.580057 + 0.814576i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q - 4 q^{2} + 36 q^{4} + 24 q^{8} - 32 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(323\)	\(346\)	\(442\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{3}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.643746	+	1.85998i	−0.455197	+	1.31521i	0.451066	+	0.892491i	\(0.351044\pi\)
							−0.906263	+	0.422715i	\(0.861077\pi\)
\(3\)	−0.458227	−	0.888835i	−0.264557	−	0.513169i
							\(5\)	2.63722	−	0.251824i	1.17940	−	0.112619i	0.513118	−	0.858318i	\(-0.328490\pi\)
0.666283	+	0.745699i	\(0.267884\pi\)
\(7\)	−0.893381	−	2.49036i	−0.337666	−	0.941266i
							\(11\)	0.479300	−	0.165887i	0.144514	−	0.0500169i	−0.253853	−	0.967243i	\(-0.581698\pi\)
0.398368	+	0.917226i	\(0.369577\pi\)
\(13\)	−0.303180	−	1.03254i	−0.0840871	−	0.286374i	0.906705	−	0.421765i	\(-0.138589\pi\)
							−0.990792	+	0.135391i	\(0.956771\pi\)
\(17\)	4.52847	+	1.81293i	1.09832	+	0.439699i	0.848823	−	0.528677i	\(-0.177312\pi\)
							0.249493	+	0.968377i	\(0.419736\pi\)
\(19\)	6.47139	−	2.59075i	1.48464	−	0.594360i	0.519122	−	0.854700i	\(-0.326259\pi\)
							0.965517	+	0.260340i	\(0.0838348\pi\)
\(23\)	3.71218	−	3.03640i	0.774042	−	0.633134i
							\(29\)	−1.13793	−	7.91448i	−0.211308	−	1.46968i	−0.768795	−	0.639496i	\(-0.779143\pi\)
0.557486	−	0.830186i	\(-0.311766\pi\)
\(31\)	−4.86736	+	0.231861i	−0.874204	+	0.0416435i	−0.479886	−	0.877331i	\(-0.659322\pi\)
							−0.394318	+	0.918974i	\(0.629019\pi\)
\(37\)	7.84350	+	5.58533i	1.28946	+	0.918223i	0.999249	−	0.0387391i	\(-0.0123341\pi\)
							0.290214	+	0.956962i	\(0.406274\pi\)
\(41\)	8.01544	+	3.66053i	1.25180	+	0.571679i	0.927340	−	0.374220i	\(-0.122090\pi\)
							0.324462	+	0.945899i	\(0.394817\pi\)
\(43\)	−6.39782	+	9.95520i	−0.975658	+	1.51815i	−0.125187	+	0.992133i	\(0.539953\pi\)
							−0.850472	+	0.526021i	\(0.823683\pi\)
\(47\)	3.75135	−	2.16584i	0.547191	−	0.315921i	−0.200797	−	0.979633i	\(-0.564353\pi\)
							0.747988	+	0.663712i	\(0.231020\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.bf.a.10.6	✓	640
7.5	odd	6	inner	483.2.bf.a.355.27	yes	640
23.7	odd	22	inner	483.2.bf.a.283.27	yes	640
161.145	even	66	inner	483.2.bf.a.145.6	yes	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.bf.a.10.6	✓	640	1.1	even	1	trivial
483.2.bf.a.145.6	yes	640	161.145	even	66	inner
483.2.bf.a.283.27	yes	640	23.7	odd	22	inner
483.2.bf.a.355.27	yes	640	7.5	odd	6	inner