Embedded newform 483.2.y.a.163.2

• Label: 483.2.y.a.163.2
• Level: $483$
• Weight: $2$
• Character: 483.163
• 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			163.2
Character	\(\chi\)	\(=\)	483.163
Dual form			483.2.y.a.403.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.95688 + 1.00884i) q^{2} +(0.786053 - 0.618159i) q^{3} +(1.65151 - 2.31922i) q^{4} +(1.06589 - 1.01632i) q^{5} +(-0.914588 + 2.00267i) q^{6} +(-2.60630 + 0.455190i) q^{7} +(-0.265434 + 1.84614i) q^{8} +(0.235759 - 0.971812i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q + 2 q^{2} - 16 q^{3} + 18 q^{4} - 2 q^{5} - 18 q^{6} + 2 q^{7} - 12 q^{8} + 16 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.95688	+	1.00884i	−1.38373	+	0.713360i	−0.979527	−	0.201312i	\(-0.935480\pi\)
							−0.404198	+	0.914672i	\(0.632449\pi\)
\(3\)	0.786053	−	0.618159i	0.453828	−	0.356894i
							\(5\)	1.06589	−	1.01632i	0.476679	−	0.454512i	−0.413246	−	0.910619i	\(-0.635605\pi\)
0.889925	+	0.456107i	\(0.150757\pi\)
\(7\)	−2.60630	+	0.455190i	−0.985089	+	0.172046i
							\(11\)	−4.31064	−	2.22229i	−1.29971	−	0.670045i	−0.337233	−	0.941421i	\(-0.609491\pi\)
−0.962473	+	0.271376i	\(0.912521\pi\)
\(13\)	0.00937255	+	0.0108165i	0.00259948	+	0.00299996i	0.757048	−	0.653359i	\(-0.226641\pi\)
							−0.754448	+	0.656359i	\(0.772096\pi\)
\(17\)	−6.37732	+	0.608960i	−1.54673	+	0.147695i	−0.833215	−	0.552949i	\(-0.813502\pi\)
							−0.713512	+	0.700643i	\(0.752896\pi\)
\(19\)	2.21412	+	0.211423i	0.507953	+	0.0485037i	0.345887	−	0.938276i	\(-0.387578\pi\)
							0.162066	+	0.986780i	\(0.448184\pi\)
\(23\)	−3.89893	+	2.79256i	−0.812982	+	0.582288i
							\(29\)	−2.06997	+	4.53261i	−0.384385	+	0.841685i	0.614233	+	0.789125i	\(0.289465\pi\)
−0.998618	+	0.0525603i	\(0.983262\pi\)
\(31\)	−1.79625	+	0.719111i	−0.322617	+	0.129156i	−0.527317	−	0.849668i	\(-0.676802\pi\)
							0.204701	+	0.978825i	\(0.434378\pi\)
\(37\)	−1.71760	+	7.08003i	−0.282371	+	1.16395i	0.637096	+	0.770784i	\(0.280135\pi\)
							−0.919468	+	0.393166i	\(0.871380\pi\)
\(41\)	0.552530	+	0.162238i	0.0862908	+	0.0253373i	0.324593	−	0.945854i	\(-0.394773\pi\)
							−0.238302	+	0.971191i	\(0.576591\pi\)
\(43\)	−0.926149	−	6.44151i	−0.141236	−	0.982321i	−0.929983	−	0.367602i	\(-0.880179\pi\)
							0.788747	−	0.614718i	\(-0.210730\pi\)
\(47\)	4.26423	−	7.38586i	0.622002	−	1.07734i	−0.367111	−	0.930177i	\(-0.619653\pi\)
							0.989112	−	0.147161i	\(-0.0470137\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.y.a.163.2	yes	320
7.4	even	3	inner	483.2.y.a.25.15	✓	320
23.12	even	11	inner	483.2.y.a.58.15	yes	320
161.81	even	33	inner	483.2.y.a.403.2	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.y.a.25.15	✓	320	7.4	even	3	inner
483.2.y.a.58.15	yes	320	23.12	even	11	inner
483.2.y.a.163.2	yes	320	1.1	even	1	trivial
483.2.y.a.403.2	yes	320	161.81	even	33	inner