Embedded newform 483.2.y.a.4.4

• Label: 483.2.y.a.4.4
• 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			4.4
Character	\(\chi\)	\(=\)	483.4
Dual form			483.2.y.a.121.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.971087 + 1.36370i) q^{2} +(-0.235759 + 0.971812i) q^{3} +(-0.262535 - 0.758544i) q^{4} +(-0.194859 + 4.09059i) q^{5} +(-1.09632 - 1.26522i) q^{6} +(-2.20100 + 1.46819i) q^{7} +(-1.92325 - 0.564717i) q^{8} +(-0.888835 - 0.458227i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q + 2 q^{2} - 16 q^{3} + 18 q^{4} - 2 q^{5} - 18 q^{6} + 2 q^{7} - 12 q^{8} + 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.971087	+	1.36370i	−0.686662	+	0.964282i	0.313231	+	0.949677i	\(0.398589\pi\)
							−0.999893	+	0.0146053i	\(0.995351\pi\)
\(3\)	−0.235759	+	0.971812i	−0.136115	+	0.561076i
							\(5\)	−0.194859	+	4.09059i	−0.0871435	+	1.82937i	0.357788	+	0.933803i	\(0.383531\pi\)
−0.444931	+	0.895565i	\(0.646772\pi\)
\(7\)	−2.20100	+	1.46819i	−0.831900	+	0.554925i
							\(11\)	1.28205	+	1.80038i	0.386552	+	0.542836i	0.961303	−	0.275492i	\(-0.0888407\pi\)
−0.574751	+	0.818328i	\(0.694901\pi\)
\(13\)	0.137456	−	0.956027i	0.0381234	−	0.265154i	−0.961841	−	0.273610i	\(-0.911782\pi\)
							0.999964	+	0.00845535i	\(0.00269146\pi\)
\(17\)	3.29464	−	0.634990i	0.799068	−	0.154008i	0.226657	−	0.973975i	\(-0.427220\pi\)
							0.572411	+	0.819967i	\(0.306008\pi\)
\(19\)	1.18326	+	0.228054i	0.271457	+	0.0523191i	0.323163	−	0.946343i	\(-0.395254\pi\)
							−0.0517058	+	0.998662i	\(0.516466\pi\)
\(23\)	3.01194	+	3.73205i	0.628033	+	0.778186i
							\(29\)	6.28173	+	7.24951i	1.16649	+	1.34620i	0.926893	+	0.375327i	\(0.122470\pi\)
0.239596	+	0.970873i	\(0.422985\pi\)
\(31\)	4.53908	−	4.32801i	0.815244	−	0.777333i	−0.162595	−	0.986693i	\(-0.551986\pi\)
							0.977839	+	0.209360i	\(0.0671380\pi\)
\(37\)	3.10897	+	1.60279i	0.511112	+	0.263496i	0.694434	−	0.719557i	\(-0.255655\pi\)
							−0.183322	+	0.983053i	\(0.558685\pi\)
\(41\)	−3.94544	−	2.53558i	−0.616174	−	0.395991i	0.194994	−	0.980804i	\(-0.437531\pi\)
							−0.811168	+	0.584814i	\(0.801168\pi\)
\(43\)	−9.25730	+	2.71819i	−1.41173	+	0.414520i	−0.896694	−	0.442651i	\(-0.854038\pi\)
							−0.515031	+	0.857171i	\(0.672220\pi\)
\(47\)	−5.28389	−	9.15197i	−0.770735	−	1.33495i	−0.937161	−	0.348898i	\(-0.886556\pi\)
							0.166426	−	0.986054i	\(-0.446777\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.y.a.4.4	✓	320
7.2	even	3	inner	483.2.y.a.142.13	yes	320
23.6	even	11	inner	483.2.y.a.466.13	yes	320
161.121	even	33	inner	483.2.y.a.121.4	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.y.a.4.4	✓	320	1.1	even	1	trivial
483.2.y.a.121.4	yes	320	161.121	even	33	inner
483.2.y.a.142.13	yes	320	7.2	even	3	inner
483.2.y.a.466.13	yes	320	23.6	even	11	inner