Embedded newform 483.2.bf.a.10.8

• Label: 483.2.bf.a.10.8
• 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			10.8
Character	\(\chi\)	\(=\)	483.10
Dual form			483.2.bf.a.145.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.544288 + 1.57262i) q^{2} +(-0.458227 - 0.888835i) q^{3} +(-0.604770 - 0.475596i) q^{4} +(1.92390 - 0.183711i) q^{5} +(1.64721 - 0.236832i) q^{6} +(0.763953 + 2.53306i) q^{7} +(-1.72283 + 1.10720i) q^{8} +(-0.580057 + 0.814576i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q - 4 q^{2} + 36 q^{4} + 24 q^{8} - 32 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.544288	+	1.57262i	−0.384870	+	1.11201i	0.571689	+	0.820470i	\(0.306288\pi\)
							−0.956559	+	0.291538i	\(0.905833\pi\)
\(3\)	−0.458227	−	0.888835i	−0.264557	−	0.513169i
							\(5\)	1.92390	−	0.183711i	0.860396	−	0.0821579i	0.344476	−	0.938795i	\(-0.388056\pi\)
0.515919	+	0.856637i	\(0.327450\pi\)
\(7\)	0.763953	+	2.53306i	0.288747	+	0.957405i
							\(11\)	3.76860	−	1.30433i	1.13628	−	0.393269i	0.306780	−	0.951780i	\(-0.400748\pi\)
0.829497	+	0.558511i	\(0.188627\pi\)
\(13\)	−0.663405	−	2.25935i	−0.183996	−	0.626631i	−0.998894	−	0.0470270i	\(-0.985025\pi\)
							0.814898	−	0.579604i	\(-0.196793\pi\)
\(17\)	3.90548	+	1.56352i	0.947217	+	0.379209i	0.793270	−	0.608870i	\(-0.208377\pi\)
							0.153948	+	0.988079i	\(0.450801\pi\)
\(19\)	−1.75029	+	0.700711i	−0.401544	+	0.160754i	−0.563639	−	0.826021i	\(-0.690599\pi\)
							0.162095	+	0.986775i	\(0.448175\pi\)
\(23\)	1.51901	+	4.54891i	0.316735	+	0.948514i
							\(29\)	1.49969	+	10.4306i	0.278486	+	1.93691i	0.343789	+	0.939047i	\(0.388290\pi\)
−0.0653026	+	0.997866i	\(0.520801\pi\)
\(31\)	7.19690	−	0.342831i	1.29260	−	0.0615742i	0.610022	−	0.792385i	\(-0.291161\pi\)
							0.682580	+	0.730810i	\(0.260858\pi\)
\(37\)	−8.71711	−	6.20742i	−1.43308	−	1.02049i	−0.992429	−	0.122817i	\(-0.960807\pi\)
							−0.440654	−	0.897677i	\(-0.645253\pi\)
\(41\)	2.91207	+	1.32990i	0.454788	+	0.207695i	0.629620	−	0.776904i	\(-0.283211\pi\)
							−0.174831	+	0.984598i	\(0.555938\pi\)
\(43\)	1.00556	−	1.56468i	0.153347	−	0.238612i	−0.756077	−	0.654482i	\(-0.772887\pi\)
							0.909424	+	0.415870i	\(0.136523\pi\)
\(47\)	−5.05862	+	2.92060i	−0.737876	+	0.426013i	−0.821297	−	0.570502i	\(-0.806749\pi\)
							0.0834205	+	0.996514i	\(0.473416\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.bf.a.10.8	✓	640
7.5	odd	6	inner	483.2.bf.a.355.25	yes	640
23.7	odd	22	inner	483.2.bf.a.283.25	yes	640
161.145	even	66	inner	483.2.bf.a.145.8	yes	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.bf.a.10.8	✓	640	1.1	even	1	trivial
483.2.bf.a.145.8	yes	640	161.145	even	66	inner
483.2.bf.a.283.25	yes	640	23.7	odd	22	inner
483.2.bf.a.355.25	yes	640	7.5	odd	6	inner