Embedded newform 483.2.y.b.4.8

• Label: 483.2.y.b.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.b.121.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.111942 + 0.157200i) q^{2} +(0.235759 - 0.971812i) q^{3} +(0.641955 + 1.85481i) q^{4} +(-0.171106 + 3.59197i) q^{5} +(0.126378 + 0.145848i) q^{6} +(-2.30883 - 1.29202i) q^{7} +(-0.733771 - 0.215455i) q^{8} +(-0.888835 - 0.458227i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q + 2 q^{2} + 16 q^{3} + 18 q^{4} - 2 q^{5} + 18 q^{6} + 2 q^{7} - 12 q^{8} + 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.111942	+	0.157200i	−0.0791548	+	0.111157i	−0.852255	−	0.523126i	\(-0.824766\pi\)
							0.773100	+	0.634284i	\(0.218705\pi\)
\(3\)	0.235759	−	0.971812i	0.136115	−	0.561076i
							\(5\)	−0.171106	+	3.59197i	−0.0765211	+	1.60638i	0.555561	+	0.831476i	\(0.312503\pi\)
−0.632083	+	0.774901i	\(0.717800\pi\)
\(7\)	−2.30883	−	1.29202i	−0.872655	−	0.488337i
							\(11\)	1.58074	+	2.21984i	0.476611	+	0.669306i	0.980964	−	0.194191i	\(-0.0622083\pi\)
−0.504353	+	0.863498i	\(0.668269\pi\)
\(13\)	0.00476466	−	0.0331389i	0.00132148	−	0.00919109i	−0.989150	−	0.146909i	\(-0.953068\pi\)
							0.990472	+	0.137718i	\(0.0439767\pi\)
\(17\)	−7.18554	+	1.38490i	−1.74275	+	0.335887i	−0.958615	−	0.284705i	\(-0.908105\pi\)
							−0.784134	+	0.620592i	\(0.786892\pi\)
\(19\)	1.20057	+	0.231392i	0.275431	+	0.0530849i	0.325096	−	0.945681i	\(-0.394603\pi\)
							−0.0496654	+	0.998766i	\(0.515815\pi\)
\(23\)	1.86902	−	4.41664i	0.389719	−	0.920934i
							\(29\)	6.71327	+	7.74752i	1.24662	+	1.43868i	0.855062	+	0.518526i	\(0.173519\pi\)
0.391561	+	0.920152i	\(0.371935\pi\)
\(31\)	−2.29403	+	2.18735i	−0.412020	+	0.392860i	−0.867542	−	0.497364i	\(-0.834301\pi\)
							0.455522	+	0.890225i	\(0.349453\pi\)
\(37\)	−8.07604	−	4.16349i	−1.32769	−	0.684473i	−0.359140	−	0.933284i	\(-0.616930\pi\)
							−0.968552	+	0.248811i	\(0.919960\pi\)
\(41\)	8.26094	+	5.30898i	1.29014	+	0.829124i	0.992103	−	0.125429i	\(-0.0400309\pi\)
							0.298040	+	0.954553i	\(0.403667\pi\)
\(43\)	1.84414	−	0.541489i	0.281229	−	0.0825763i	−0.138076	−	0.990422i	\(-0.544092\pi\)
							0.419305	+	0.907845i	\(0.362274\pi\)
\(47\)	4.78237	+	8.28331i	0.697580	+	1.20824i	0.969303	+	0.245869i	\(0.0790734\pi\)
							−0.271723	+	0.962376i	\(0.587593\pi\)

(See \(a_n\) instead)

Twists

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

    

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