Embedded newform 483.2.bf.a.388.13

• Label: 483.2.bf.a.388.13
• 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			388.13
Character	\(\chi\)	\(=\)	483.388
Dual form			483.2.bf.a.61.13

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

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

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)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.bf.a.61.13	✓	640	161.61	even	66	inner
483.2.bf.a.199.20	yes	640	23.15	odd	22	inner
483.2.bf.a.250.20	yes	640	7.5	odd	6	inner
483.2.bf.a.388.13	yes	640	1.1	even	1	trivial