Embedded newform 483.2.y.b.4.11

• Label: 483.2.y.b.4.11
• 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			4.11
Character	\(\chi\)	\(=\)	483.4
Dual form			483.2.y.b.121.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.733120 - 1.02952i) q^{2} +(0.235759 - 0.971812i) q^{3} +(0.131683 + 0.380473i) q^{4} +(0.00357914 - 0.0751355i) q^{5} +(-0.827663 - 0.955174i) q^{6} +(1.84836 + 1.89303i) q^{7} +(2.91361 + 0.855512i) q^{8} +(-0.888835 - 0.458227i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q + 2 q^{2} + 16 q^{3} + 18 q^{4} - 2 q^{5} + 18 q^{6} + 2 q^{7} - 12 q^{8} + 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.733120	−	1.02952i	0.518394	−	0.727983i	−0.469531	−	0.882916i	\(-0.655577\pi\)
							0.987925	+	0.154933i	\(0.0495163\pi\)
\(3\)	0.235759	−	0.971812i	0.136115	−	0.561076i
							\(5\)	0.00357914	−	0.0751355i	0.00160064	−	0.0336016i	−0.997926	−	0.0643760i	\(-0.979494\pi\)
0.999526	+	0.0307744i	\(0.00979734\pi\)
\(7\)	1.84836	+	1.89303i	0.698614	+	0.715499i
							\(11\)	3.67794	+	5.16495i	1.10894	+	1.55729i	0.793730	+	0.608270i	\(0.208136\pi\)
0.315212	+	0.949021i	\(0.397924\pi\)
\(13\)	0.444495	−	3.09153i	0.123281	−	0.857436i	−0.830518	−	0.556991i	\(-0.811956\pi\)
							0.953799	−	0.300445i	\(-0.0971352\pi\)
\(17\)	−6.91553	+	1.33286i	−1.67726	+	0.323266i	−0.937073	−	0.349133i	\(-0.886476\pi\)
							−0.740189	+	0.672399i	\(0.765264\pi\)
\(19\)	−0.610815	−	0.117725i	−0.140131	−	0.0270079i	0.118704	−	0.992930i	\(-0.462126\pi\)
							−0.258834	+	0.965922i	\(0.583338\pi\)
\(23\)	−1.04556	−	4.68047i	−0.218014	−	0.975946i
							\(29\)	−6.63857	−	7.66131i	−1.23275	−	1.42267i	−0.871639	−	0.490148i	\(-0.836943\pi\)
−0.361111	−	0.932523i	\(-0.617603\pi\)
\(31\)	3.91029	−	3.72846i	0.702310	−	0.669651i	−0.252253	−	0.967661i	\(-0.581172\pi\)
							0.954563	+	0.298011i	\(0.0963231\pi\)
\(37\)	0.313056	+	0.161392i	0.0514662	+	0.0265327i	0.483768	−	0.875197i	\(-0.339268\pi\)
							−0.432301	+	0.901729i	\(0.642298\pi\)
\(41\)	−6.85357	−	4.40452i	−1.07035	−	0.687871i	−0.118040	−	0.993009i	\(-0.537661\pi\)
							−0.952308	+	0.305138i	\(0.901297\pi\)
\(43\)	−4.84728	+	1.42329i	−0.739203	+	0.217049i	−0.629593	−	0.776925i	\(-0.716778\pi\)
							−0.109610	+	0.993975i	\(0.534960\pi\)
\(47\)	2.30886	+	3.99906i	0.336782	+	0.583324i	0.983825	−	0.179130i	\(-0.0573281\pi\)
							−0.647043	+	0.762453i	\(0.723995\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.y.b.4.11	✓	320
7.2	even	3	inner	483.2.y.b.142.6	yes	320
23.6	even	11	inner	483.2.y.b.466.6	yes	320
161.121	even	33	inner	483.2.y.b.121.11	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.y.b.4.11	✓	320	1.1	even	1	trivial
483.2.y.b.121.11	yes	320	161.121	even	33	inner
483.2.y.b.142.6	yes	320	7.2	even	3	inner
483.2.y.b.466.6	yes	320	23.6	even	11	inner