Embedded newform 483.2.y.a.4.6

• Label: 483.2.y.a.4.6
• 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			4.6
Character	\(\chi\)	\(=\)	483.4
Dual form			483.2.y.a.121.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.851765 + 1.19614i) q^{2} +(-0.235759 + 0.971812i) q^{3} +(-0.0511029 - 0.147652i) q^{4} +(0.117190 - 2.46013i) q^{5} +(-0.961608 - 1.10975i) q^{6} +(-2.08452 - 1.62935i) q^{7} +(-2.59773 - 0.762762i) q^{8} +(-0.888835 - 0.458227i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q + 2 q^{2} - 16 q^{3} + 18 q^{4} - 2 q^{5} - 18 q^{6} + 2 q^{7} - 12 q^{8} + 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.851765	+	1.19614i	−0.602289	+	0.845796i	−0.997342	−	0.0728621i	\(-0.976787\pi\)
							0.395053	+	0.918658i	\(0.370726\pi\)
\(3\)	−0.235759	+	0.971812i	−0.136115	+	0.561076i
							\(5\)	0.117190	−	2.46013i	0.0524091	−	1.10020i	−0.807537	−	0.589817i	\(-0.799200\pi\)
0.859946	−	0.510385i	\(-0.170497\pi\)
\(7\)	−2.08452	−	1.62935i	−0.787875	−	0.615835i
							\(11\)	2.66626	+	3.74424i	0.803909	+	1.12893i	0.989247	+	0.146257i	\(0.0467227\pi\)
−0.185338	+	0.982675i	\(0.559338\pi\)
\(13\)	0.969180	−	6.74079i	0.268802	−	1.86956i	−0.191073	−	0.981576i	\(-0.561197\pi\)
							0.459875	−	0.887984i	\(-0.347894\pi\)
\(17\)	0.693635	−	0.133687i	0.168231	−	0.0324239i	−0.104440	−	0.994531i	\(-0.533305\pi\)
							0.272671	+	0.962107i	\(0.412093\pi\)
\(19\)	5.51725	+	1.06336i	1.26574	+	0.243952i	0.777571	−	0.628795i	\(-0.216452\pi\)
							0.488172	+	0.872747i	\(0.337664\pi\)
\(23\)	−3.35487	−	3.42708i	−0.699538	−	0.714595i
							\(29\)	−2.24396	−	2.58967i	−0.416693	−	0.480890i	0.508133	−	0.861278i	\(-0.330336\pi\)
−0.924827	+	0.380389i	\(0.875790\pi\)
\(31\)	5.05287	−	4.81790i	0.907523	−	0.865321i	−0.0838387	−	0.996479i	\(-0.526718\pi\)
							0.991361	+	0.131158i	\(0.0418696\pi\)
\(37\)	−2.31964	−	1.19586i	−0.381346	−	0.196598i	0.256886	−	0.966442i	\(-0.417304\pi\)
							−0.638232	+	0.769844i	\(0.720334\pi\)
\(41\)	−1.97956	−	1.27218i	−0.309155	−	0.198682i	0.376858	−	0.926271i	\(-0.377004\pi\)
							−0.686013	+	0.727589i	\(0.740641\pi\)
\(43\)	4.40504	−	1.29344i	0.671762	−	0.197247i	0.0719743	−	0.997406i	\(-0.477070\pi\)
							0.599787	+	0.800159i	\(0.295252\pi\)
\(47\)	5.59851	+	9.69690i	0.816627	+	1.41444i	0.908154	+	0.418636i	\(0.137492\pi\)
							−0.0915274	+	0.995803i	\(0.529175\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.y.a.4.6	✓	320
7.2	even	3	inner	483.2.y.a.142.11	yes	320
23.6	even	11	inner	483.2.y.a.466.11	yes	320
161.121	even	33	inner	483.2.y.a.121.6	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.y.a.4.6	✓	320	1.1	even	1	trivial
483.2.y.a.121.6	yes	320	161.121	even	33	inner
483.2.y.a.142.11	yes	320	7.2	even	3	inner
483.2.y.a.466.11	yes	320	23.6	even	11	inner