Embedded newform 483.2.y.b.4.10

• Label: 483.2.y.b.4.10
• 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			4.10
Character	\(\chi\)	\(=\)	483.4
Dual form			483.2.y.b.121.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.199911 - 0.280735i) q^{2} +(0.235759 - 0.971812i) q^{3} +(0.615288 + 1.77776i) q^{4} +(-0.0359296 + 0.754254i) q^{5} +(-0.225691 - 0.260461i) q^{6} +(2.50738 - 0.844407i) q^{7} +(1.28344 + 0.376852i) q^{8} +(-0.888835 - 0.458227i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q + 2 q^{2} + 16 q^{3} + 18 q^{4} - 2 q^{5} + 18 q^{6} + 2 q^{7} - 12 q^{8} + 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.199911	−	0.280735i	0.141358	−	0.198510i	−0.737827	−	0.674990i	\(-0.764148\pi\)
							0.879185	+	0.476480i	\(0.158087\pi\)
\(3\)	0.235759	−	0.971812i	0.136115	−	0.561076i
							\(5\)	−0.0359296	+	0.754254i	−0.0160682	+	0.337313i	0.976492	+	0.215555i	\(0.0691561\pi\)
−0.992560	+	0.121758i	\(0.961147\pi\)
\(7\)	2.50738	−	0.844407i	0.947702	−	0.319156i
							\(11\)	−0.469273	−	0.659002i	−0.141491	−	0.198697i	0.737749	−	0.675075i	\(-0.235889\pi\)
−0.879240	+	0.476378i	\(0.841949\pi\)
\(13\)	−0.698049	+	4.85504i	−0.193604	+	1.34655i	0.628767	+	0.777594i	\(0.283560\pi\)
							−0.822371	+	0.568952i	\(0.807349\pi\)
\(17\)	5.29061	−	1.01968i	1.28316	−	0.247309i	0.498313	−	0.866997i	\(-0.333953\pi\)
							0.784848	+	0.619688i	\(0.212741\pi\)
\(19\)	−0.866568	−	0.167017i	−0.198804	−	0.0383164i	0.0888769	−	0.996043i	\(-0.471672\pi\)
							−0.287681	+	0.957726i	\(0.592884\pi\)
\(23\)	0.502495	−	4.76943i	0.104778	−	0.994496i
							\(29\)	2.97137	+	3.42914i	0.551769	+	0.636775i	0.961294	−	0.275523i	\(-0.0888511\pi\)
−0.409525	+	0.912299i	\(0.634306\pi\)
\(31\)	−3.93497	+	3.75198i	−0.706741	+	0.673876i	−0.955601	−	0.294664i	\(-0.904792\pi\)
							0.248860	+	0.968539i	\(0.419944\pi\)
\(37\)	−0.990899	−	0.510844i	−0.162903	−	0.0839823i	0.374847	−	0.927087i	\(-0.377695\pi\)
							−0.537750	+	0.843104i	\(0.680726\pi\)
\(41\)	−8.08277	−	5.19448i	−1.26232	−	0.811241i	−0.273717	−	0.961810i	\(-0.588253\pi\)
							−0.988599	+	0.150569i	\(0.951889\pi\)
\(43\)	9.49137	−	2.78692i	1.44742	−	0.425001i	0.538733	−	0.842477i	\(-0.318903\pi\)
							0.908688	+	0.417476i	\(0.137085\pi\)
\(47\)	−6.00086	−	10.3938i	−0.875315	−	1.51609i	−0.856426	−	0.516269i	\(-0.827320\pi\)
							−0.0188889	−	0.999822i	\(-0.506013\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.y.b.4.10	✓	320
7.2	even	3	inner	483.2.y.b.142.7	yes	320
23.6	even	11	inner	483.2.y.b.466.7	yes	320
161.121	even	33	inner	483.2.y.b.121.10	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.y.b.4.10	✓	320	1.1	even	1	trivial
483.2.y.b.121.10	yes	320	161.121	even	33	inner
483.2.y.b.142.7	yes	320	7.2	even	3	inner
483.2.y.b.466.7	yes	320	23.6	even	11	inner