Embedded newform 483.2.y.b.331.5

• Label: 483.2.y.b.331.5
• 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			331.5
Character	\(\chi\)	\(=\)	483.331
Dual form			483.2.y.b.340.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.356781 + 1.47067i) q^{2} +(-0.327068 + 0.945001i) q^{3} +(-0.257912 - 0.132963i) q^{4} +(-2.60174 - 1.04158i) q^{5} +(-1.27309 - 0.818168i) q^{6} +(-1.21008 - 2.35281i) q^{7} +(-1.69448 + 1.95553i) q^{8} +(-0.786053 - 0.618159i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q + 2 q^{2} + 16 q^{3} + 18 q^{4} - 2 q^{5} + 18 q^{6} + 2 q^{7} - 12 q^{8} + 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.356781	+	1.47067i	−0.252282	+	1.03992i	0.694869	+	0.719136i	\(0.255462\pi\)
							−0.947152	+	0.320786i	\(0.896053\pi\)
\(3\)	−0.327068	+	0.945001i	−0.188833	+	0.545596i
							\(5\)	−2.60174	−	1.04158i	−1.16353	−	0.465809i	−0.292131	−	0.956378i	\(-0.594364\pi\)
−0.871402	+	0.490570i	\(0.836789\pi\)
\(7\)	−1.21008	−	2.35281i	−0.457366	−	0.889279i
							\(11\)	0.502644	+	2.07193i	0.151553	+	0.624709i	0.996022	+	0.0891054i	\(0.0284008\pi\)
−0.844469	+	0.535604i	\(0.820084\pi\)
\(13\)	−0.526237	−	1.15230i	−0.145952	−	0.319590i	0.822510	−	0.568750i	\(-0.192573\pi\)
							−0.968462	+	0.249160i	\(0.919846\pi\)
\(17\)	0.0871720	−	1.82996i	0.0211423	−	0.443832i	−0.963333	−	0.268309i	\(-0.913535\pi\)
							0.984475	−	0.175523i	\(-0.0561617\pi\)
\(19\)	−0.325233	−	6.82747i	−0.0746135	−	1.56633i	−0.657298	−	0.753631i	\(-0.728301\pi\)
							0.582685	−	0.812698i	\(-0.302002\pi\)
\(23\)	4.76909	−	0.505765i	0.994424	−	0.105459i
							\(29\)	−6.31168	−	4.05627i	−1.17205	−	0.753231i	−0.198142	−	0.980173i	\(-0.563491\pi\)
−0.973908	+	0.226942i	\(0.927127\pi\)
\(31\)	−4.86378	−	0.937417i	−0.873561	−	0.168365i	−0.267283	−	0.963618i	\(-0.586126\pi\)
							−0.606278	+	0.795253i	\(0.707338\pi\)
\(37\)	6.22977	+	4.89915i	1.02417	+	0.805415i	0.981069	−	0.193656i	\(-0.0620346\pi\)
							0.0430986	+	0.999071i	\(0.486277\pi\)
\(41\)	−0.570516	−	3.96802i	−0.0890996	−	0.619701i	−0.984624	−	0.174688i	\(-0.944108\pi\)
							0.895524	−	0.445013i	\(-0.146801\pi\)
\(43\)	−2.40572	−	2.77634i	−0.366868	−	0.423389i	0.542061	−	0.840339i	\(-0.317644\pi\)
							−0.908929	+	0.416951i	\(0.863099\pi\)
\(47\)	−5.93354	+	10.2772i	−0.865496	+	1.49908i	0.00105773	+	0.999999i	\(0.499663\pi\)
							−0.866554	+	0.499084i	\(0.833670\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.y.b.331.5	yes	320
7.4	even	3	inner	483.2.y.b.193.5	✓	320
23.18	even	11	inner	483.2.y.b.478.5	yes	320
161.18	even	33	inner	483.2.y.b.340.5	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.y.b.193.5	✓	320	7.4	even	3	inner
483.2.y.b.331.5	yes	320	1.1	even	1	trivial
483.2.y.b.340.5	yes	320	161.18	even	33	inner
483.2.y.b.478.5	yes	320	23.18	even	11	inner