Embedded newform 483.2.bf.a.388.3

• Label: 483.2.bf.a.388.3
• 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			388.3
Character	\(\chi\)	\(=\)	483.388
Dual form			483.2.bf.a.61.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.96199 + 1.54292i) q^{2} +(0.814576 - 0.580057i) q^{3} +(0.997264 - 4.11078i) q^{4} +(2.33608 + 0.450242i) q^{5} +(-0.703204 + 2.39489i) q^{6} +(-0.792487 - 2.52427i) q^{7} +(2.31225 + 5.06312i) q^{8} +(0.327068 - 0.945001i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q - 4 q^{2} + 36 q^{4} + 24 q^{8} - 32 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.96199	+	1.54292i	−1.38734	+	1.09101i	−0.402904	+	0.915242i	\(0.631999\pi\)
							−0.984431	+	0.175770i	\(0.943758\pi\)
\(3\)	0.814576	−	0.580057i	0.470296	−	0.334896i
							\(5\)	2.33608	+	0.450242i	1.04473	+	0.201354i	0.682613	−	0.730780i	\(-0.260844\pi\)
0.362112	+	0.932134i	\(0.382056\pi\)
\(7\)	−0.792487	−	2.52427i	−0.299532	−	0.954086i
							\(11\)	3.31117	−	4.21049i	0.998355	−	1.26951i	0.0355342	−	0.999368i	\(-0.488687\pi\)
0.962820	−	0.270143i	\(-0.0870708\pi\)
\(13\)	−1.53274	+	2.38499i	−0.425105	+	0.661477i	−0.986064	−	0.166367i	\(-0.946796\pi\)
							0.560958	+	0.827844i	\(0.310433\pi\)
\(17\)	−2.09340	+	1.99605i	−0.507723	+	0.484113i	−0.900095	−	0.435694i	\(-0.856503\pi\)
							0.392372	+	0.919807i	\(0.371655\pi\)
\(19\)	−5.86014	−	5.58763i	−1.34441	−	1.28189i	−0.927935	−	0.372743i	\(-0.878417\pi\)
							−0.416473	−	0.909148i	\(-0.636734\pi\)
\(23\)	2.72136	−	3.94895i	0.567442	−	0.823413i
							\(29\)	−2.12886	−	0.625088i	−0.395318	−	0.116076i	0.0780305	−	0.996951i	\(-0.475137\pi\)
−0.473349	+	0.880875i	\(0.656955\pi\)
\(31\)	0.694028	−	7.26819i	0.124651	−	1.30541i	−0.689301	−	0.724475i	\(-0.742082\pi\)
							0.813952	−	0.580931i	\(-0.197312\pi\)
\(37\)	2.18048	+	0.754670i	0.358468	+	0.124067i	0.500365	−	0.865814i	\(-0.333199\pi\)
							−0.141897	+	0.989881i	\(0.545320\pi\)
\(41\)	4.41413	+	3.82487i	0.689372	+	0.597344i	0.927470	−	0.373897i	\(-0.121979\pi\)
							−0.238099	+	0.971241i	\(0.576524\pi\)
\(43\)	6.34454	+	2.89746i	0.967534	+	0.441858i	0.835562	−	0.549396i	\(-0.185142\pi\)
							0.131972	+	0.991254i	\(0.457869\pi\)
\(47\)	−6.81760	+	3.93614i	−0.994450	+	0.574146i	−0.906601	−	0.421988i	\(-0.861332\pi\)
							−0.0878482	+	0.996134i	\(0.527999\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.bf.a.388.3	yes	640
7.5	odd	6	inner	483.2.bf.a.250.30	yes	640
23.15	odd	22	inner	483.2.bf.a.199.30	yes	640
161.61	even	66	inner	483.2.bf.a.61.3	✓	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.bf.a.61.3	✓	640	161.61	even	66	inner
483.2.bf.a.199.30	yes	640	23.15	odd	22	inner
483.2.bf.a.250.30	yes	640	7.5	odd	6	inner
483.2.bf.a.388.3	yes	640	1.1	even	1	trivial