Embedded newform 483.2.y.a.4.16

• Label: 483.2.y.a.4.16
• 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			4.16
Character	\(\chi\)	\(=\)	483.4
Dual form			483.2.y.a.121.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.55491 - 2.18356i) q^{2} +(-0.235759 + 0.971812i) q^{3} +(-1.69606 - 4.90045i) q^{4} +(0.00530352 - 0.111335i) q^{5} +(1.75543 + 2.02587i) q^{6} +(-2.64473 - 0.0735467i) q^{7} +(-8.19360 - 2.40586i) q^{8} +(-0.888835 - 0.458227i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q + 2 q^{2} - 16 q^{3} + 18 q^{4} - 2 q^{5} - 18 q^{6} + 2 q^{7} - 12 q^{8} + 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.55491	−	2.18356i	1.09948	−	1.54401i	0.289852	−	0.957071i	\(-0.406394\pi\)
							0.809632	−	0.586938i	\(-0.199667\pi\)
\(3\)	−0.235759	+	0.971812i	−0.136115	+	0.561076i
							\(5\)	0.00530352	−	0.111335i	0.00237181	−	0.0497904i	−0.997371	−	0.0724623i	\(-0.976914\pi\)
0.999743	+	0.0226719i	\(0.00721732\pi\)
\(7\)	−2.64473	−	0.0735467i	−0.999614	−	0.0277981i
							\(11\)	−2.32482	−	3.26475i	−0.700959	−	0.984360i	−0.999551	−	0.0299784i	\(-0.990456\pi\)
0.298591	−	0.954381i	\(-0.403483\pi\)
\(13\)	0.512016	−	3.56115i	0.142008	−	0.987686i	−0.786822	−	0.617180i	\(-0.788275\pi\)
							0.928830	−	0.370506i	\(-0.120816\pi\)
\(17\)	6.54393	−	1.26124i	1.58714	−	0.305895i	0.682109	−	0.731251i	\(-0.261063\pi\)
							0.905026	+	0.425355i	\(0.139851\pi\)
\(19\)	−3.49080	−	0.672796i	−0.800844	−	0.154350i	−0.227621	−	0.973750i	\(-0.573095\pi\)
							−0.573223	+	0.819400i	\(0.694307\pi\)
\(23\)	0.337865	+	4.78392i	0.0704497	+	0.997515i
							\(29\)	−3.35176	−	3.86814i	−0.622407	−	0.718296i	0.353755	−	0.935338i	\(-0.384905\pi\)
−0.976162	+	0.217042i	\(0.930359\pi\)
\(31\)	4.58879	−	4.37540i	0.824170	−	0.785845i	−0.155207	−	0.987882i	\(-0.549605\pi\)
							0.979378	+	0.202037i	\(0.0647561\pi\)
\(37\)	5.48132	+	2.82582i	0.901124	+	0.464562i	0.845552	−	0.533893i	\(-0.179271\pi\)
							0.0555719	+	0.998455i	\(0.482302\pi\)
\(41\)	8.17303	+	5.25248i	1.27641	+	0.820300i	0.990441	−	0.137938i	\(-0.0440474\pi\)
							0.285971	+	0.958238i	\(0.407684\pi\)
\(43\)	−1.67206	+	0.490962i	−0.254987	+	0.0748710i	−0.406728	−	0.913549i	\(-0.633330\pi\)
							0.151741	+	0.988420i	\(0.451512\pi\)
\(47\)	3.17457	+	5.49852i	0.463059	+	0.802041i	0.999112	−	0.0421430i	\(-0.0134185\pi\)
							−0.536053	+	0.844185i	\(0.680085\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.y.a.4.16	✓	320
7.2	even	3	inner	483.2.y.a.142.1	yes	320
23.6	even	11	inner	483.2.y.a.466.1	yes	320
161.121	even	33	inner	483.2.y.a.121.16	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.y.a.4.16	✓	320	1.1	even	1	trivial
483.2.y.a.121.16	yes	320	161.121	even	33	inner
483.2.y.a.142.1	yes	320	7.2	even	3	inner
483.2.y.a.466.1	yes	320	23.6	even	11	inner