Embedded newform 483.2.y.a.4.5

• Label: 483.2.y.a.4.5
• 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			4.5
Character	\(\chi\)	\(=\)	483.4
Dual form			483.2.y.a.121.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.957403 + 1.34448i) q^{2} +(-0.235759 + 0.971812i) q^{3} +(-0.236882 - 0.684426i) q^{4} +(0.115570 - 2.42610i) q^{5} +(-1.08087 - 1.24739i) q^{6} +(2.21964 - 1.43986i) q^{7} +(-2.02036 - 0.593231i) q^{8} +(-0.888835 - 0.458227i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q + 2 q^{2} - 16 q^{3} + 18 q^{4} - 2 q^{5} - 18 q^{6} + 2 q^{7} - 12 q^{8} + 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.957403	+	1.34448i	−0.676986	+	0.950694i	0.323005	+	0.946397i	\(0.395307\pi\)
							−0.999991	+	0.00429672i	\(0.998632\pi\)
\(3\)	−0.235759	+	0.971812i	−0.136115	+	0.561076i
							\(5\)	0.115570	−	2.42610i	0.0516843	−	1.08499i	−0.812869	−	0.582446i	\(-0.802096\pi\)
0.864554	−	0.502541i	\(-0.167601\pi\)
\(7\)	2.21964	−	1.43986i	0.838945	−	0.544216i
							\(11\)	−0.243478	−	0.341917i	−0.0734114	−	0.103092i	0.776238	−	0.630440i	\(-0.217126\pi\)
−0.849649	+	0.527348i	\(0.823186\pi\)
\(13\)	−0.871137	+	6.05889i	−0.241610	+	1.68043i	0.402437	+	0.915448i	\(0.368163\pi\)
							−0.644047	+	0.764986i	\(0.722746\pi\)
\(17\)	5.69918	−	1.09843i	1.38225	−	0.266408i	0.556692	−	0.830719i	\(-0.312070\pi\)
							0.825562	+	0.564311i	\(0.190858\pi\)
\(19\)	4.77961	+	0.921194i	1.09652	+	0.211336i	0.705263	−	0.708946i	\(-0.250829\pi\)
							0.391255	+	0.920282i	\(0.372041\pi\)
\(23\)	3.96736	+	2.69445i	0.827252	+	0.561831i
							\(29\)	−0.408146	−	0.471026i	−0.0757909	−	0.0874673i	0.716588	−	0.697497i	\(-0.245703\pi\)
−0.792379	+	0.610030i	\(0.791158\pi\)
\(31\)	−5.22627	+	4.98324i	−0.938667	+	0.895017i	−0.994638	−	0.103416i	\(-0.967023\pi\)
							0.0559716	+	0.998432i	\(0.482174\pi\)
\(37\)	6.81960	+	3.51575i	1.12114	+	0.577986i	0.916173	−	0.400784i	\(-0.131262\pi\)
							0.204963	+	0.978770i	\(0.434292\pi\)
\(41\)	3.15401	+	2.02696i	0.492574	+	0.316558i	0.763240	−	0.646115i	\(-0.223607\pi\)
							−0.270666	+	0.962673i	\(0.587244\pi\)
\(43\)	−0.780351	+	0.229132i	−0.119002	+	0.0349422i	−0.340691	−	0.940175i	\(-0.610661\pi\)
							0.221689	+	0.975117i	\(0.428843\pi\)
\(47\)	−4.23304	−	7.33184i	−0.617452	−	1.06946i	−0.989949	−	0.141425i	\(-0.954831\pi\)
							0.372496	−	0.928034i	\(-0.378502\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.y.a.4.5	✓	320
7.2	even	3	inner	483.2.y.a.142.12	yes	320
23.6	even	11	inner	483.2.y.a.466.12	yes	320
161.121	even	33	inner	483.2.y.a.121.5	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.y.a.4.5	✓	320	1.1	even	1	trivial
483.2.y.a.121.5	yes	320	161.121	even	33	inner
483.2.y.a.142.12	yes	320	7.2	even	3	inner
483.2.y.a.466.12	yes	320	23.6	even	11	inner