Embedded newform 483.2.bf.a.10.3

• Label: 483.2.bf.a.10.3
• 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			10.3
Character	\(\chi\)	\(=\)	483.10
Dual form			483.2.bf.a.145.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.838936 + 2.42395i) q^{2} +(-0.458227 - 0.888835i) q^{3} +(-3.59960 - 2.83076i) q^{4} +(-2.60640 + 0.248881i) q^{5} +(2.53891 - 0.365041i) q^{6} +(2.49779 - 0.872377i) q^{7} +(5.56577 - 3.57691i) q^{8} +(-0.580057 + 0.814576i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q - 4 q^{2} + 36 q^{4} + 24 q^{8} - 32 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.838936	+	2.42395i	−0.593218	+	1.71399i	0.101620	+	0.994823i	\(0.467597\pi\)
							−0.694838	+	0.719166i	\(0.744524\pi\)
\(3\)	−0.458227	−	0.888835i	−0.264557	−	0.513169i
							\(5\)	−2.60640	+	0.248881i	−1.16562	+	0.111303i	−0.659870	−	0.751380i	\(-0.729389\pi\)
−0.505746	+	0.862682i	\(0.668783\pi\)
\(7\)	2.49779	−	0.872377i	0.944076	−	0.329727i
							\(11\)	0.681421	−	0.235842i	0.205456	−	0.0711090i	−0.222402	−	0.974955i	\(-0.571390\pi\)
0.427858	+	0.903846i	\(0.359268\pi\)
\(13\)	0.235885	+	0.803350i	0.0654227	+	0.222809i	0.985715	−	0.168422i	\(-0.0538673\pi\)
							−0.920292	+	0.391232i	\(0.872049\pi\)
\(17\)	3.50053	+	1.40140i	0.849004	+	0.339890i	0.755035	−	0.655685i	\(-0.227620\pi\)
							0.0939698	+	0.995575i	\(0.470044\pi\)
\(19\)	3.75940	−	1.50504i	0.862466	−	0.345279i	0.102100	−	0.994774i	\(-0.467444\pi\)
							0.760366	+	0.649495i	\(0.225020\pi\)
\(23\)	−3.97216	+	2.68737i	−0.828252	+	0.560356i
							\(29\)	1.43212	+	9.96059i	0.265937	+	1.84963i	0.485757	+	0.874094i	\(0.338544\pi\)
−0.219820	+	0.975541i	\(0.570547\pi\)
\(31\)	−6.94196	+	0.330686i	−1.24681	+	0.0593930i	−0.660583	−	0.750753i	\(-0.729691\pi\)
							−0.586229	+	0.810146i	\(0.699388\pi\)
\(37\)	9.19638	+	6.54871i	1.51188	+	1.07660i	0.972035	+	0.234834i	\(0.0754548\pi\)
							0.539840	+	0.841767i	\(0.318485\pi\)
\(41\)	3.47102	+	1.58516i	0.542083	+	0.247561i	0.667587	−	0.744532i	\(-0.267327\pi\)
							−0.125504	+	0.992093i	\(0.540055\pi\)
\(43\)	3.07710	−	4.78806i	0.469254	−	0.730173i	−0.523278	−	0.852162i	\(-0.675291\pi\)
							0.992531	+	0.121989i	\(0.0389273\pi\)
\(47\)	4.30124	−	2.48332i	0.627401	−	0.362230i	−0.152344	−	0.988328i	\(-0.548682\pi\)
							0.779745	+	0.626097i	\(0.215349\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.bf.a.10.3	✓	640
7.5	odd	6	inner	483.2.bf.a.355.30	yes	640
23.7	odd	22	inner	483.2.bf.a.283.30	yes	640
161.145	even	66	inner	483.2.bf.a.145.3	yes	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.bf.a.10.3	✓	640	1.1	even	1	trivial
483.2.bf.a.145.3	yes	640	161.145	even	66	inner
483.2.bf.a.283.30	yes	640	23.7	odd	22	inner
483.2.bf.a.355.30	yes	640	7.5	odd	6	inner