Embedded newform 483.2.y.b.331.11

• Label: 483.2.y.b.331.11
• 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			331.11
Character	\(\chi\)	\(=\)	483.331
Dual form			483.2.y.b.340.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.267514 - 1.10271i) q^{2} +(-0.327068 + 0.945001i) q^{3} +(0.633270 + 0.326473i) q^{4} +(1.59984 + 0.640478i) q^{5} +(0.954565 + 0.613461i) q^{6} +(0.433383 - 2.61002i) q^{7} +(2.01555 - 2.32606i) q^{8} +(-0.786053 - 0.618159i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q + 2 q^{2} + 16 q^{3} + 18 q^{4} - 2 q^{5} + 18 q^{6} + 2 q^{7} - 12 q^{8} + 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.267514	−	1.10271i	0.189161	−	0.779732i	−0.795600	−	0.605822i	\(-0.792844\pi\)
							0.984761	−	0.173911i	\(-0.0556404\pi\)
\(3\)	−0.327068	+	0.945001i	−0.188833	+	0.545596i
							\(5\)	1.59984	+	0.640478i	0.715469	+	0.286431i	0.700696	−	0.713460i	\(-0.252873\pi\)
0.0147734	+	0.999891i	\(0.495297\pi\)
\(7\)	0.433383	−	2.61002i	0.163803	−	0.986493i
							\(11\)	0.546530	+	2.25283i	0.164785	+	0.679253i	0.992926	+	0.118734i	\(0.0378834\pi\)
−0.828141	+	0.560520i	\(0.810601\pi\)
\(13\)	−0.181622	−	0.397697i	−0.0503730	−	0.110301i	0.882772	−	0.469801i	\(-0.155675\pi\)
							−0.933145	+	0.359500i	\(0.882947\pi\)
\(17\)	0.0171063	−	0.359106i	0.00414890	−	0.0870960i	−0.995843	−	0.0910847i	\(-0.970967\pi\)
							0.999992	+	0.00398866i	\(0.00126963\pi\)
\(19\)	0.208867	+	4.38466i	0.0479174	+	1.00591i	0.886960	+	0.461845i	\(0.152813\pi\)
							−0.839043	+	0.544065i	\(0.816884\pi\)
\(23\)	1.92830	−	4.39109i	0.402078	−	0.915605i
							\(29\)	8.72573	+	5.60769i	1.62033	+	1.04132i	0.955783	+	0.294074i	\(0.0950111\pi\)
0.664545	+	0.747248i	\(0.268625\pi\)
\(31\)	−2.85559	−	0.550369i	−0.512879	−	0.0988493i	−0.0737559	−	0.997276i	\(-0.523499\pi\)
							−0.439123	+	0.898427i	\(0.644711\pi\)
\(37\)	0.0360387	+	0.0283412i	0.00592473	+	0.00465926i	0.621117	−	0.783718i	\(-0.286679\pi\)
							−0.615192	+	0.788377i	\(0.710922\pi\)
\(41\)	0.876300	+	6.09480i	0.136855	+	0.951848i	0.936323	+	0.351140i	\(0.114206\pi\)
							−0.799468	+	0.600709i	\(0.794885\pi\)
\(43\)	−6.63969	−	7.66261i	−1.01254	−	1.16854i	−0.985633	−	0.168900i	\(-0.945979\pi\)
							−0.0269101	−	0.999638i	\(-0.508567\pi\)
\(47\)	−2.87323	+	4.97658i	−0.419103	+	0.725908i	−0.995850	−	0.0910151i	\(-0.970989\pi\)
							0.576746	+	0.816923i	\(0.304322\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.y.b.331.11	yes	320
7.4	even	3	inner	483.2.y.b.193.11	✓	320
23.18	even	11	inner	483.2.y.b.478.11	yes	320
161.18	even	33	inner	483.2.y.b.340.11	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.y.b.193.11	✓	320	7.4	even	3	inner
483.2.y.b.331.11	yes	320	1.1	even	1	trivial
483.2.y.b.340.11	yes	320	161.18	even	33	inner
483.2.y.b.478.11	yes	320	23.18	even	11	inner