Embedded newform 483.2.y.b.478.5

• Label: 483.2.y.b.478.5
• Level: $483$
• Weight: $2$
• Character: 483.478
• 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			478.5
Character	\(\chi\)	\(=\)	483.478
Dual form			483.2.y.b.193.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.09525 + 1.04432i) q^{2} +(0.981929 + 0.189251i) q^{3} +(0.0138068 - 0.289839i) q^{4} +(2.20290 + 1.73238i) q^{5} +(-1.27309 + 0.818168i) q^{6} +(2.57057 + 0.626249i) q^{7} +(-1.69448 - 1.95553i) q^{8} +(0.928368 + 0.371662i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q + 2 q^{2} + 16 q^{3} + 18 q^{4} - 2 q^{5} + 18 q^{6} + 2 q^{7} - 12 q^{8} + 16 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(323\)	\(346\)	\(442\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{6}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.09525	+	1.04432i	−0.774458	+	0.738444i	−0.970225	−	0.242207i	\(-0.922129\pi\)
							0.195767	+	0.980650i	\(0.437280\pi\)
\(3\)	0.981929	+	0.189251i	0.566917	+	0.109264i
							\(5\)	2.20290	+	1.73238i	0.985169	+	0.774745i	0.974314	−	0.225196i	\(-0.0723022\pi\)
0.0108552	+	0.999941i	\(0.496545\pi\)
\(7\)	2.57057	+	0.626249i	0.971583	+	0.236700i
							\(11\)	1.54302	+	1.47127i	0.465238	+	0.443603i	0.886081	−	0.463530i	\(-0.153417\pi\)
−0.420844	+	0.907133i	\(0.638266\pi\)
\(13\)	−0.526237	+	1.15230i	−0.145952	+	0.319590i	−0.968462	−	0.249160i	\(-0.919846\pi\)
							0.822510	+	0.568750i	\(0.192573\pi\)
\(17\)	−1.62838	−	0.839489i	−0.394941	−	0.203606i	0.249304	−	0.968425i	\(-0.419798\pi\)
							−0.644245	+	0.764819i	\(0.722828\pi\)
\(19\)	6.07538	−	3.13208i	1.39379	−	0.718547i	0.412475	−	0.910969i	\(-0.364664\pi\)
							0.981312	+	0.192421i	\(0.0616340\pi\)
\(23\)	−1.94654	−	4.38303i	−0.405881	−	0.913926i
							\(29\)	−6.31168	+	4.05627i	−1.17205	+	0.753231i	−0.973908	−	0.226942i	\(-0.927127\pi\)
−0.198142	+	0.980173i	\(0.563491\pi\)
\(31\)	1.62006	−	4.68087i	0.290972	−	0.840709i	−0.700876	−	0.713283i	\(-0.747207\pi\)
							0.991848	−	0.127426i	\(-0.0406714\pi\)
\(37\)	−7.35767	−	2.94557i	−1.20959	−	0.484248i	−0.322824	−	0.946459i	\(-0.604632\pi\)
							−0.886770	+	0.462211i	\(0.847056\pi\)
\(41\)	−0.570516	+	3.96802i	−0.0890996	+	0.619701i	0.895524	+	0.445013i	\(0.146801\pi\)
							−0.984624	+	0.174688i	\(0.944108\pi\)
\(43\)	−2.40572	+	2.77634i	−0.366868	+	0.423389i	−0.908929	−	0.416951i	\(-0.863099\pi\)
							0.542061	+	0.840339i	\(0.317644\pi\)
\(47\)	−5.93354	+	10.2772i	−0.865496	+	1.49908i	0.00105773	+	0.999999i	\(0.499663\pi\)
							−0.866554	+	0.499084i	\(0.833670\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.y.b.478.5	yes	320
7.4	even	3	inner	483.2.y.b.340.5	yes	320
23.9	even	11	inner	483.2.y.b.331.5	yes	320
161.32	even	33	inner	483.2.y.b.193.5	✓	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.y.b.193.5	✓	320	161.32	even	33	inner
483.2.y.b.331.5	yes	320	23.9	even	11	inner
483.2.y.b.340.5	yes	320	7.4	even	3	inner
483.2.y.b.478.5	yes	320	1.1	even	1	trivial