Embedded newform 483.2.bf.a.10.18

• Label: 483.2.bf.a.10.18
• 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			10.18
Character	\(\chi\)	\(=\)	483.10
Dual form			483.2.bf.a.145.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0503654 - 0.145521i) q^{2} +(-0.458227 - 0.888835i) q^{3} +(1.55347 + 1.22166i) q^{4} +(-3.87542 + 0.370058i) q^{5} +(-0.152423 + 0.0219152i) q^{6} +(2.22141 - 1.43713i) q^{7} +(0.515109 - 0.331040i) q^{8} +(-0.580057 + 0.814576i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q - 4 q^{2} + 36 q^{4} + 24 q^{8} - 32 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0503654	−	0.145521i	0.0356137	−	0.102899i	−0.925792	−	0.378032i	\(-0.876601\pi\)
							0.961406	+	0.275133i	\(0.0887220\pi\)
\(3\)	−0.458227	−	0.888835i	−0.264557	−	0.513169i
							\(5\)	−3.87542	+	0.370058i	−1.73314	+	0.165495i	−0.913352	−	0.407172i	\(-0.866515\pi\)
−0.819788	+	0.572666i	\(0.805909\pi\)
\(7\)	2.22141	−	1.43713i	0.839614	−	0.543184i
							\(11\)	3.50372	−	1.21265i	1.05641	−	0.365628i	0.257000	−	0.966411i	\(-0.417266\pi\)
0.799412	+	0.600784i	\(0.205145\pi\)
\(13\)	−1.12996	−	3.84828i	−0.313394	−	1.06732i	−0.954088	−	0.299526i	\(-0.903172\pi\)
							0.640694	−	0.767796i	\(-0.278647\pi\)
\(17\)	−0.758418	−	0.303625i	−0.183943	−	0.0736398i	0.277862	−	0.960621i	\(-0.410374\pi\)
							−0.461806	+	0.886981i	\(0.652798\pi\)
\(19\)	5.11541	−	2.04790i	1.17356	−	0.469821i	0.298768	−	0.954326i	\(-0.403424\pi\)
							0.874789	+	0.484505i	\(0.161000\pi\)
\(23\)	3.94696	−	2.72424i	0.822999	−	0.568043i
							\(29\)	−0.161705	−	1.12468i	−0.0300278	−	0.208848i	0.969284	−	0.245945i	\(-0.0790982\pi\)
−0.999312	+	0.0370966i	\(0.988189\pi\)
\(31\)	7.28204	−	0.346886i	1.30789	−	0.0623026i	0.617979	−	0.786194i	\(-0.287952\pi\)
							0.689914	+	0.723892i	\(0.257649\pi\)
\(37\)	−7.73994	−	5.51159i	−1.27244	−	0.906099i	−0.273987	−	0.961733i	\(-0.588343\pi\)
							−0.998451	+	0.0556338i	\(0.982282\pi\)
\(41\)	−2.88222	−	1.31627i	−0.450127	−	0.205566i	0.177435	−	0.984132i	\(-0.443220\pi\)
							−0.627562	+	0.778566i	\(0.715947\pi\)
\(43\)	−5.19518	+	8.08386i	−0.792257	+	1.23278i	0.176386	+	0.984321i	\(0.443559\pi\)
							−0.968643	+	0.248455i	\(0.920077\pi\)
\(47\)	−1.73198	+	0.999959i	−0.252635	+	0.145859i	−0.620970	−	0.783834i	\(-0.713261\pi\)
							0.368335	+	0.929693i	\(0.379928\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.bf.a.10.18	✓	640
7.5	odd	6	inner	483.2.bf.a.355.15	yes	640
23.7	odd	22	inner	483.2.bf.a.283.15	yes	640
161.145	even	66	inner	483.2.bf.a.145.18	yes	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.bf.a.10.18	✓	640	1.1	even	1	trivial
483.2.bf.a.145.18	yes	640	161.145	even	66	inner
483.2.bf.a.283.15	yes	640	23.7	odd	22	inner
483.2.bf.a.355.15	yes	640	7.5	odd	6	inner