Embedded newform 483.2.y.b.4.14

• Label: 483.2.y.b.4.14
• Level: $483$
• Weight: $2$
• Character: 483.4
• Analytic conductor: $3.857$
• Analytic rank: $0$
• Dimension: $320$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 483 = 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	483.y (of order \(33\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.85677441763\)
Analytic rank:	\(0\)
Dimension:	\(320\)
Relative dimension:	\(16\) over \(\Q(\zeta_{33})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{33}]$

Embedding invariants

Embedding label			4.14
Character	\(\chi\)	\(=\)	483.4
Dual form			483.2.y.b.121.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.20076 - 1.68624i) q^{2} +(0.235759 - 0.971812i) q^{3} +(-0.747425 - 2.15954i) q^{4} +(-0.0741223 + 1.55602i) q^{5} +(-1.35561 - 1.56446i) q^{6} +(2.38094 - 1.15374i) q^{7} +(-0.566529 - 0.166348i) q^{8} +(-0.888835 - 0.458227i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q + 2 q^{2} + 16 q^{3} + 18 q^{4} - 2 q^{5} + 18 q^{6} + 2 q^{7} - 12 q^{8} + 16 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/483\mathbb{Z}\right)^\times\).

\(n\)	\(323\)	\(346\)	\(442\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{2}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.20076	−	1.68624i	0.849068	−	1.19235i	−0.130542	−	0.991443i	\(-0.541672\pi\)
							0.979610	−	0.200907i	\(-0.0643888\pi\)
\(3\)	0.235759	−	0.971812i	0.136115	−	0.561076i
							\(5\)	−0.0741223	+	1.55602i	−0.0331485	+	0.695873i	0.919736	+	0.392538i	\(0.128403\pi\)
−0.952884	+	0.303334i	\(0.901900\pi\)
\(7\)	2.38094	−	1.15374i	0.899912	−	0.436072i
							\(11\)	−1.77811	−	2.49700i	−0.536120	−	0.752875i	0.454318	−	0.890840i	\(-0.349883\pi\)
−0.990437	+	0.137965i	\(0.955944\pi\)
\(13\)	0.522701	−	3.63546i	0.144971	−	1.00830i	−0.779325	−	0.626620i	\(-0.784438\pi\)
							0.924296	−	0.381676i	\(-0.124653\pi\)
\(17\)	−2.78028	+	0.535856i	−0.674318	+	0.129964i	−0.514904	−	0.857248i	\(-0.672173\pi\)
							−0.159414	+	0.987212i	\(0.550960\pi\)
\(19\)	−2.94669	−	0.567929i	−0.676018	−	0.130292i	−0.160324	−	0.987064i	\(-0.551254\pi\)
							−0.515694	+	0.856773i	\(0.672466\pi\)
\(23\)	2.76202	+	3.92062i	0.575921	+	0.817505i
							\(29\)	4.38596	+	5.06167i	0.814452	+	0.939928i	0.999080	−	0.0428808i	\(-0.0136536\pi\)
−0.184628	+	0.982808i	\(0.559108\pi\)
\(31\)	−6.78641	+	6.47083i	−1.21888	+	1.16220i	−0.237236	+	0.971452i	\(0.576242\pi\)
							−0.981640	+	0.190744i	\(0.938910\pi\)
\(37\)	−1.88068	−	0.969558i	−0.309182	−	0.159394i	0.296652	−	0.954986i	\(-0.404130\pi\)
							−0.605834	+	0.795591i	\(0.707160\pi\)
\(41\)	4.01016	+	2.57717i	0.626281	+	0.402487i	0.814931	−	0.579558i	\(-0.196775\pi\)
							−0.188650	+	0.982044i	\(0.560411\pi\)
\(43\)	2.31170	−	0.678776i	0.352531	−	0.103512i	−0.100672	−	0.994920i	\(-0.532099\pi\)
							0.453203	+	0.891407i	\(0.350281\pi\)
\(47\)	5.81201	+	10.0667i	0.847769	+	1.46838i	0.883194	+	0.469007i	\(0.155388\pi\)
							−0.0354249	+	0.999372i	\(0.511278\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.y.b.4.14	✓	320
7.2	even	3	inner	483.2.y.b.142.3	yes	320
23.6	even	11	inner	483.2.y.b.466.3	yes	320
161.121	even	33	inner	483.2.y.b.121.14	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.y.b.4.14	✓	320	1.1	even	1	trivial
483.2.y.b.121.14	yes	320	161.121	even	33	inner
483.2.y.b.142.3	yes	320	7.2	even	3	inner
483.2.y.b.466.3	yes	320	23.6	even	11	inner