Embedded newform 483.2.bf.a.61.13

• Label: 483.2.bf.a.61.13
• 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			61.13
Character	\(\chi\)	\(=\)	483.61
Dual form			483.2.bf.a.388.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.546266 - 0.429588i) q^{2} +(-0.814576 - 0.580057i) q^{3} +(-0.357657 - 1.47428i) q^{4} +(1.83567 - 0.353796i) q^{5} +(0.195789 + 0.666798i) q^{6} +(0.230543 + 2.63569i) q^{7} +(-1.01534 + 2.22329i) q^{8} +(0.327068 + 0.945001i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q - 4 q^{2} + 36 q^{4} + 24 q^{8} - 32 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.546266	−	0.429588i	−0.386268	−	0.303765i	0.406075	−	0.913840i	\(-0.366897\pi\)
							−0.792344	+	0.610075i	\(0.791139\pi\)
\(3\)	−0.814576	−	0.580057i	−0.470296	−	0.334896i
							\(5\)	1.83567	−	0.353796i	0.820935	−	0.158222i	0.238544	−	0.971132i	\(-0.423330\pi\)
0.582390	+	0.812909i	\(0.302118\pi\)
\(7\)	0.230543	+	2.63569i	0.0871370	+	0.996196i
							\(11\)	3.07670	+	3.91235i	0.927661	+	1.17962i	0.983451	+	0.181174i	\(0.0579899\pi\)
−0.0557900	+	0.998443i	\(0.517768\pi\)
\(13\)	3.49893	+	5.44445i	0.970430	+	1.51002i	0.856253	+	0.516557i	\(0.172787\pi\)
							0.114177	+	0.993460i	\(0.463577\pi\)
\(17\)	−1.49686	−	1.42726i	−0.363043	−	0.346161i	0.486500	−	0.873681i	\(-0.338274\pi\)
							−0.849543	+	0.527520i	\(0.823122\pi\)
\(19\)	3.74090	−	3.56695i	0.858222	−	0.818313i	−0.126586	−	0.991956i	\(-0.540402\pi\)
							0.984809	+	0.173642i	\(0.0555536\pi\)
\(23\)	1.51441	−	4.55045i	0.315775	−	0.948834i
							\(29\)	−8.23550	+	2.41816i	−1.52929	+	0.449041i	−0.934835	−	0.355084i	\(-0.884452\pi\)
−0.594460	+	0.804125i	\(0.702634\pi\)
\(31\)	0.368791	+	3.86215i	0.0662368	+	0.693663i	0.966217	+	0.257732i	\(0.0829750\pi\)
							−0.899980	+	0.435932i	\(0.856419\pi\)
\(37\)	9.74973	−	3.37441i	1.60285	−	0.554750i	0.628112	−	0.778123i	\(-0.283828\pi\)
							0.974733	+	0.223373i	\(0.0717067\pi\)
\(41\)	−5.54206	+	4.80222i	−0.865524	+	0.749981i	−0.969628	−	0.244586i	\(-0.921348\pi\)
							0.104104	+	0.994566i	\(0.466803\pi\)
\(43\)	8.96222	−	4.09291i	1.36673	−	0.624162i	0.409182	−	0.912453i	\(-0.365814\pi\)
							0.957543	+	0.288290i	\(0.0930868\pi\)
\(47\)	−3.79868	−	2.19317i	−0.554095	−	0.319907i	0.196677	−	0.980468i	\(-0.436985\pi\)
							−0.750772	+	0.660561i	\(0.770318\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.bf.a.61.13	✓	640
7.3	odd	6	inner	483.2.bf.a.199.20	yes	640
23.20	odd	22	inner	483.2.bf.a.250.20	yes	640
161.66	even	66	inner	483.2.bf.a.388.13	yes	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.bf.a.61.13	✓	640	1.1	even	1	trivial
483.2.bf.a.199.20	yes	640	7.3	odd	6	inner
483.2.bf.a.250.20	yes	640	23.20	odd	22	inner
483.2.bf.a.388.13	yes	640	161.66	even	66	inner