Embedded newform 483.2.bf.a.145.3

• Label: 483.2.bf.a.145.3
• Level: $483$
• Weight: $2$
• Character: 483.145
• 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			145.3
Character	\(\chi\)	\(=\)	483.145
Dual form			483.2.bf.a.10.3

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

\(n\)	\(323\)	\(346\)	\(442\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{19}{22}\right)\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.bf.a.10.3	✓	640	161.10	even	66	inner
483.2.bf.a.145.3	yes	640	1.1	even	1	trivial
483.2.bf.a.283.30	yes	640	7.3	odd	6	inner
483.2.bf.a.355.30	yes	640	23.10	odd	22	inner