Embedded newform 483.2.bf.a.388.31

• Label: 483.2.bf.a.388.31
• Level: $483$
• Weight: $2$
• Character: 483.388
• 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			388.31
Character	\(\chi\)	\(=\)	483.388
Dual form			483.2.bf.a.61.31

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.10417 - 1.65474i) q^{2} +(0.814576 - 0.580057i) q^{3} +(1.21785 - 5.02007i) q^{4} +(1.23922 + 0.238839i) q^{5} +(0.754164 - 2.56845i) q^{6} +(0.245101 + 2.63437i) q^{7} +(-3.52028 - 7.70834i) q^{8} +(0.327068 - 0.945001i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q - 4 q^{2} + 36 q^{4} + 24 q^{8} - 32 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.10417	−	1.65474i	1.48787	−	1.17008i	0.544308	−	0.838886i	\(-0.316792\pi\)
							0.943564	−	0.331190i	\(-0.107450\pi\)
\(3\)	0.814576	−	0.580057i	0.470296	−	0.334896i
							\(5\)	1.23922	+	0.238839i	0.554194	+	0.106812i	0.458656	−	0.888614i	\(-0.348331\pi\)
0.0955376	+	0.995426i	\(0.469543\pi\)
\(7\)	0.245101	+	2.63437i	0.0926393	+	0.995700i
							\(11\)	−1.73542	+	2.20677i	−0.523249	+	0.665365i	−0.973806	−	0.227383i	\(-0.926983\pi\)
0.450556	+	0.892748i	\(0.351226\pi\)
\(13\)	−0.367617	+	0.572023i	−0.101959	+	0.158651i	−0.888485	−	0.458907i	\(-0.848241\pi\)
							0.786526	+	0.617557i	\(0.211878\pi\)
\(17\)	−4.01664	+	3.82986i	−0.974178	+	0.928877i	−0.997487	−	0.0708512i	\(-0.977428\pi\)
							0.0233086	+	0.999728i	\(0.492580\pi\)
\(19\)	3.02905	+	2.88819i	0.694912	+	0.662597i	0.952807	−	0.303576i	\(-0.0981805\pi\)
							−0.257896	+	0.966173i	\(0.583029\pi\)
\(23\)	−4.52876	−	1.57807i	−0.944312	−	0.329050i
							\(29\)	5.17706	+	1.52012i	0.961356	+	0.282280i	0.724508	−	0.689267i	\(-0.242067\pi\)
0.236849	+	0.971547i	\(0.423885\pi\)
\(31\)	0.648880	−	6.79538i	0.116542	−	1.22049i	−0.728220	−	0.685344i	\(-0.759652\pi\)
							0.844762	−	0.535142i	\(-0.179742\pi\)
\(37\)	−8.22632	−	2.84716i	−1.35240	−	0.468070i	−0.447724	−	0.894172i	\(-0.647765\pi\)
							−0.904675	+	0.426102i	\(0.859887\pi\)
\(41\)	4.29246	+	3.71944i	0.670370	+	0.580879i	0.922116	−	0.386915i	\(-0.126459\pi\)
							−0.251746	+	0.967793i	\(0.581005\pi\)
\(43\)	2.19834	+	1.00395i	0.335243	+	0.153100i	0.575923	−	0.817504i	\(-0.304643\pi\)
							−0.240679	+	0.970605i	\(0.577370\pi\)
\(47\)	6.70803	−	3.87289i	0.978467	−	0.564918i	0.0766603	−	0.997057i	\(-0.475574\pi\)
							0.901807	+	0.432139i	\(0.142241\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.bf.a.388.31	yes	640
7.5	odd	6	inner	483.2.bf.a.250.2	yes	640
23.15	odd	22	inner	483.2.bf.a.199.2	yes	640
161.61	even	66	inner	483.2.bf.a.61.31	✓	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.bf.a.61.31	✓	640	161.61	even	66	inner
483.2.bf.a.199.2	yes	640	23.15	odd	22	inner
483.2.bf.a.250.2	yes	640	7.5	odd	6	inner
483.2.bf.a.388.31	yes	640	1.1	even	1	trivial