Embedded newform 483.2.y.b.331.3

• Label: 483.2.y.b.331.3
• Level: $483$
• Weight: $2$
• Character: 483.331
• 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			331.3
Character	\(\chi\)	\(=\)	483.331
Dual form			483.2.y.b.340.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.479413 + 1.97617i) q^{2} +(-0.327068 + 0.945001i) q^{3} +(-1.89773 - 0.978346i) q^{4} +(3.28882 + 1.31665i) q^{5} +(-1.71068 - 1.09939i) q^{6} +(0.889851 - 2.49162i) q^{7} +(0.179868 - 0.207579i) q^{8} +(-0.786053 - 0.618159i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q + 2 q^{2} + 16 q^{3} + 18 q^{4} - 2 q^{5} + 18 q^{6} + 2 q^{7} - 12 q^{8} + 16 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(323\)	\(346\)	\(442\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{5}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.479413	+	1.97617i	−0.338996	+	1.39736i	0.505975	+	0.862548i	\(0.331133\pi\)
							−0.844971	+	0.534813i	\(0.820382\pi\)
\(3\)	−0.327068	+	0.945001i	−0.188833	+	0.545596i
							\(5\)	3.28882	+	1.31665i	1.47081	+	0.588822i	0.962241	−	0.272201i	\(-0.0877515\pi\)
0.508566	+	0.861023i	\(0.330176\pi\)
\(7\)	0.889851	−	2.49162i	0.336332	−	0.941744i
							\(11\)	0.938413	+	3.86819i	0.282942	+	1.16630i	0.918872	+	0.394557i	\(0.129102\pi\)
−0.635929	+	0.771747i	\(0.719383\pi\)
\(13\)	−0.232566	−	0.509248i	−0.0645021	−	0.141240i	0.874635	−	0.484781i	\(-0.161101\pi\)
							−0.939138	+	0.343541i	\(0.888373\pi\)
\(17\)	−0.0693264	+	1.45534i	−0.0168141	+	0.352972i	0.974745	+	0.223319i	\(0.0716892\pi\)
							−0.991559	+	0.129653i	\(0.958614\pi\)
\(19\)	0.219093	+	4.59933i	0.0502635	+	1.05516i	0.873304	+	0.487176i	\(0.161973\pi\)
							−0.823040	+	0.567983i	\(0.807724\pi\)
\(23\)	0.416605	+	4.77770i	0.0868682	+	0.996220i
							\(29\)	−5.64801	−	3.62976i	−1.04881	−	0.674029i	−0.101660	−	0.994819i	\(-0.532415\pi\)
−0.947150	+	0.320790i	\(0.896052\pi\)
\(31\)	4.49506	+	0.866352i	0.807336	+	0.155601i	0.576188	−	0.817317i	\(-0.304540\pi\)
							0.231148	+	0.972919i	\(0.425752\pi\)
\(37\)	−6.20306	−	4.87814i	−1.01978	−	0.801962i	−0.0394264	−	0.999222i	\(-0.512553\pi\)
							−0.980351	+	0.197261i	\(0.936795\pi\)
\(41\)	−0.713712	−	4.96398i	−0.111463	−	0.775243i	−0.966498	−	0.256673i	\(-0.917374\pi\)
							0.855035	−	0.518570i	\(-0.173535\pi\)
\(43\)	−0.474440	−	0.547532i	−0.0723514	−	0.0834979i	0.718420	−	0.695610i	\(-0.244866\pi\)
							−0.790771	+	0.612112i	\(0.790320\pi\)
\(47\)	4.46117	−	7.72697i	0.650728	−	1.12709i	−0.332218	−	0.943203i	\(-0.607797\pi\)
							0.982946	−	0.183892i	\(-0.0588697\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.y.b.331.3	yes	320
7.4	even	3	inner	483.2.y.b.193.3	✓	320
23.18	even	11	inner	483.2.y.b.478.3	yes	320
161.18	even	33	inner	483.2.y.b.340.3	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.y.b.193.3	✓	320	7.4	even	3	inner
483.2.y.b.331.3	yes	320	1.1	even	1	trivial
483.2.y.b.340.3	yes	320	161.18	even	33	inner
483.2.y.b.478.3	yes	320	23.18	even	11	inner