Embedded newform 483.2.y.b.16.1

• Label: 483.2.y.b.16.1
• Level: $483$
• Weight: $2$
• Character: 483.16
• 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			16.1
Character	\(\chi\)	\(=\)	483.16
Dual form			483.2.y.b.151.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.860562 - 2.48643i) q^{2} +(-0.888835 - 0.458227i) q^{3} +(-3.86967 + 3.04314i) q^{4} +(0.317869 + 0.0303528i) q^{5} +(-0.374451 + 2.60436i) q^{6} +(1.99020 + 1.74330i) q^{7} +(6.46974 + 4.15785i) q^{8} +(0.580057 + 0.814576i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q + 2 q^{2} + 16 q^{3} + 18 q^{4} - 2 q^{5} + 18 q^{6} + 2 q^{7} - 12 q^{8} + 16 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(323\)	\(346\)	\(442\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{4}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.860562	−	2.48643i	−0.608510	−	1.75817i	−0.651002	−	0.759076i	\(-0.725651\pi\)
							0.0424924	−	0.999097i	\(-0.486470\pi\)
\(3\)	−0.888835	−	0.458227i	−0.513169	−	0.264557i
							\(5\)	0.317869	+	0.0303528i	0.142155	+	0.0135742i	0.165891	−	0.986144i	\(-0.446950\pi\)
−0.0237357	+	0.999718i	\(0.507556\pi\)
\(7\)	1.99020	+	1.74330i	0.752226	+	0.658905i
							\(11\)	0.115845	−	0.334712i	0.0349286	−	0.100920i	−0.926188	−	0.377061i	\(-0.876935\pi\)
0.961117	+	0.276142i	\(0.0890559\pi\)
\(13\)	−1.39083	−	0.408383i	−0.385746	−	0.113265i	0.0831089	−	0.996540i	\(-0.473515\pi\)
							−0.468855	+	0.883275i	\(0.655333\pi\)
\(17\)	6.46244	−	2.58717i	1.56737	−	0.627481i	0.584438	−	0.811438i	\(-0.301315\pi\)
							0.982934	+	0.183957i	\(0.0588906\pi\)
\(19\)	2.23484	+	0.894694i	0.512707	+	0.205257i	0.613565	−	0.789644i	\(-0.289735\pi\)
							−0.100858	+	0.994901i	\(0.532159\pi\)
\(23\)	4.68278	+	1.03516i	0.976428	+	0.215845i
							\(29\)	0.485445	−	3.37635i	0.0901449	−	0.626972i	−0.893795	−	0.448475i	\(-0.851967\pi\)
0.983940	−	0.178497i	\(-0.0571235\pi\)
\(31\)	−0.130746	+	2.74469i	−0.0234826	+	0.492961i	0.956238	+	0.292591i	\(0.0945174\pi\)
							−0.979720	+	0.200370i	\(0.935786\pi\)
\(37\)	−1.20731	−	1.69542i	−0.198480	−	0.278726i	0.703326	−	0.710867i	\(-0.251697\pi\)
							−0.901806	+	0.432142i	\(0.857758\pi\)
\(41\)	1.63811	+	3.58695i	0.255829	+	0.560188i	0.993350	−	0.115137i	\(-0.0367307\pi\)
							−0.737520	+	0.675325i	\(0.764003\pi\)
\(43\)	5.03795	−	3.23769i	0.768280	−	0.493744i	−0.0968452	−	0.995299i	\(-0.530875\pi\)
							0.865125	+	0.501556i	\(0.167239\pi\)
\(47\)	−1.03037	+	1.78466i	−0.150295	+	0.260319i	−0.931336	−	0.364161i	\(-0.881356\pi\)
							0.781041	+	0.624480i	\(0.214689\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.y.b.16.1	✓	320
7.4	even	3	inner	483.2.y.b.361.16	yes	320
23.13	even	11	inner	483.2.y.b.289.16	yes	320
161.151	even	33	inner	483.2.y.b.151.1	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.y.b.16.1	✓	320	1.1	even	1	trivial
483.2.y.b.151.1	yes	320	161.151	even	33	inner
483.2.y.b.289.16	yes	320	23.13	even	11	inner
483.2.y.b.361.16	yes	320	7.4	even	3	inner