Embedded newform 483.2.bf.a.61.3

• Label: 483.2.bf.a.61.3
• 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			61.3
Character	\(\chi\)	\(=\)	483.61
Dual form			483.2.bf.a.388.3

$q$-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 \(\left(\mathbb{Z}/483\mathbb{Z}\right)^\times\).

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

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)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.bf.a.61.3	✓	640	1.1	even	1	trivial
483.2.bf.a.199.30	yes	640	7.3	odd	6	inner
483.2.bf.a.250.30	yes	640	23.20	odd	22	inner
483.2.bf.a.388.3	yes	640	161.66	even	66	inner