Embedded newform 483.2.y.a.4.8

• Label: 483.2.y.a.4.8
• 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			4.8
Character	\(\chi\)	\(=\)	483.4
Dual form			483.2.y.a.121.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.264798 + 0.371856i) q^{2} +(-0.235759 + 0.971812i) q^{3} +(0.585977 + 1.69307i) q^{4} +(0.162988 - 3.42155i) q^{5} +(-0.298946 - 0.345002i) q^{6} +(-2.64103 - 0.158003i) q^{7} +(-1.66077 - 0.487645i) q^{8} +(-0.888835 - 0.458227i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q + 2 q^{2} - 16 q^{3} + 18 q^{4} - 2 q^{5} - 18 q^{6} + 2 q^{7} - 12 q^{8} + 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.264798	+	0.371856i	−0.187240	+	0.262942i	−0.897516	−	0.440981i	\(-0.854631\pi\)
							0.710276	+	0.703923i	\(0.248570\pi\)
\(3\)	−0.235759	+	0.971812i	−0.136115	+	0.561076i
							\(5\)	0.162988	−	3.42155i	0.0728906	−	1.53016i	−0.605728	−	0.795672i	\(-0.707118\pi\)
0.678619	−	0.734491i	\(-0.262579\pi\)
\(7\)	−2.64103	−	0.158003i	−0.998215	−	0.0597196i
							\(11\)	−2.78188	−	3.90660i	−0.838768	−	1.17788i	−0.982128	−	0.188214i	\(-0.939730\pi\)
0.143360	−	0.989671i	\(-0.454209\pi\)
\(13\)	−0.587228	+	4.08426i	−0.162868	+	1.13277i	0.730326	+	0.683098i	\(0.239368\pi\)
							−0.893194	+	0.449671i	\(0.851541\pi\)
\(17\)	−7.65264	+	1.47493i	−1.85604	+	0.357722i	−0.989864	−	0.142021i	\(-0.954640\pi\)
							−0.866174	+	0.499743i	\(0.833428\pi\)
\(19\)	−6.79320	−	1.30928i	−1.55847	−	0.300370i	−0.664079	−	0.747662i	\(-0.731176\pi\)
							−0.894388	+	0.447292i	\(0.852388\pi\)
\(23\)	3.64381	+	3.11811i	0.759788	+	0.650171i
							\(29\)	3.38215	+	3.90321i	0.628050	+	0.724808i	0.977215	−	0.212253i	\(-0.0680802\pi\)
−0.349165	+	0.937061i	\(0.613535\pi\)
\(31\)	5.87814	−	5.60480i	1.05575	−	1.00665i	0.0557796	−	0.998443i	\(-0.482236\pi\)
							0.999966	−	0.00820854i	\(-0.00261289\pi\)
\(37\)	0.405989	+	0.209302i	0.0667442	+	0.0344090i	0.491276	−	0.871004i	\(-0.336531\pi\)
							−0.424531	+	0.905413i	\(0.639561\pi\)
\(41\)	−2.27640	−	1.46296i	−0.355515	−	0.228475i	0.350677	−	0.936497i	\(-0.385952\pi\)
							−0.706191	+	0.708021i	\(0.749588\pi\)
\(43\)	1.87696	−	0.551126i	0.286234	−	0.0840459i	−0.135464	−	0.990782i	\(-0.543252\pi\)
							0.421698	+	0.906736i	\(0.361434\pi\)
\(47\)	−1.33113	−	2.30558i	−0.194165	−	0.336304i	0.752462	−	0.658636i	\(-0.228866\pi\)
							−0.946626	+	0.322333i	\(0.895533\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.y.a.4.8	✓	320
7.2	even	3	inner	483.2.y.a.142.9	yes	320
23.6	even	11	inner	483.2.y.a.466.9	yes	320
161.121	even	33	inner	483.2.y.a.121.8	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.y.a.4.8	✓	320	1.1	even	1	trivial
483.2.y.a.121.8	yes	320	161.121	even	33	inner
483.2.y.a.142.9	yes	320	7.2	even	3	inner
483.2.y.a.466.9	yes	320	23.6	even	11	inner