Embedded newform 483.2.bf.a.10.30

• Label: 483.2.bf.a.10.30
• 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			10.30
Character	\(\chi\)	\(=\)	483.10
Dual form			483.2.bf.a.145.30

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.772442 - 2.23182i) q^{2} +(-0.458227 - 0.888835i) q^{3} +(-2.81227 - 2.21159i) q^{4} +(-0.0722748 + 0.00690140i) q^{5} +(-2.33768 + 0.336107i) q^{6} +(-2.49742 - 0.873437i) q^{7} +(-3.13459 + 2.01448i) q^{8} +(-0.580057 + 0.814576i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q - 4 q^{2} + 36 q^{4} + 24 q^{8} - 32 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.772442	−	2.23182i	0.546199	−	1.57814i	−0.246897	−	0.969042i	\(-0.579411\pi\)
							0.793096	−	0.609096i	\(-0.208468\pi\)
\(3\)	−0.458227	−	0.888835i	−0.264557	−	0.513169i
							\(5\)	−0.0722748	+	0.00690140i	−0.0323223	+	0.00308640i	−0.111205	−	0.993798i	\(-0.535471\pi\)
0.0788824	+	0.996884i	\(0.474865\pi\)
\(7\)	−2.49742	−	0.873437i	−0.943936	−	0.330128i
							\(11\)	−2.84369	+	0.984209i	−0.857404	+	0.296750i	−0.720178	−	0.693789i	\(-0.755940\pi\)
−0.137226	+	0.990540i	\(0.543819\pi\)
\(13\)	0.146486	+	0.498884i	0.0406278	+	0.138366i	0.977309	−	0.211816i	\(-0.0679379\pi\)
							−0.936682	+	0.350182i	\(0.886120\pi\)
\(17\)	−2.14126	−	0.857233i	−0.519333	−	0.207909i	0.0971619	−	0.995269i	\(-0.469024\pi\)
							−0.616495	+	0.787359i	\(0.711448\pi\)
\(19\)	2.45133	−	0.981364i	0.562373	−	0.225140i	−0.0730180	−	0.997331i	\(-0.523263\pi\)
							0.635391	+	0.772190i	\(0.280839\pi\)
\(23\)	1.82378	−	4.43552i	0.380284	−	0.924870i
							\(29\)	−0.422103	−	2.93579i	−0.0783826	−	0.545163i	−0.990740	−	0.135769i	\(-0.956649\pi\)
0.912358	−	0.409394i	\(-0.134260\pi\)
\(31\)	5.95083	−	0.283473i	1.06880	−	0.0509132i	0.494201	−	0.869348i	\(-0.335461\pi\)
							0.574600	+	0.818435i	\(0.305158\pi\)
\(37\)	−0.156240	−	0.111258i	−0.0256857	−	0.0182907i	0.567142	−	0.823620i	\(-0.308049\pi\)
							−0.592828	+	0.805329i	\(0.701988\pi\)
\(41\)	−6.38500	−	2.91593i	−0.997169	−	0.455392i	−0.151131	−	0.988514i	\(-0.548291\pi\)
							−0.846038	+	0.533122i	\(0.821019\pi\)
\(43\)	−0.950097	+	1.47838i	−0.144888	+	0.225451i	−0.906111	−	0.423040i	\(-0.860963\pi\)
							0.761223	+	0.648491i	\(0.224599\pi\)
\(47\)	10.8676	−	6.27443i	1.58521	−	0.915220i	0.591127	−	0.806579i	\(-0.298683\pi\)
							0.994081	−	0.108642i	\(-0.0346501\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.bf.a.10.30	✓	640
7.5	odd	6	inner	483.2.bf.a.355.3	yes	640
23.7	odd	22	inner	483.2.bf.a.283.3	yes	640
161.145	even	66	inner	483.2.bf.a.145.30	yes	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.bf.a.10.30	✓	640	1.1	even	1	trivial
483.2.bf.a.145.30	yes	640	161.145	even	66	inner
483.2.bf.a.283.3	yes	640	23.7	odd	22	inner
483.2.bf.a.355.3	yes	640	7.5	odd	6	inner