Embedded newform 483.2.bf.a.10.9

• Label: 483.2.bf.a.10.9
• Level: $483$
• Weight: $2$
• Character: 483.10
• Analytic conductor: $3.857$
• Analytic rank: $0$
• Dimension: $640$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 483 = 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	483.bf (of order \(66\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.85677441763\)
Analytic rank:	\(0\)
Dimension:	\(640\)
Relative dimension:	\(32\) over \(\Q(\zeta_{66})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{66}]$

Embedding invariants

Embedding label			10.9
Character	\(\chi\)	\(=\)	483.10
Dual form			483.2.bf.a.145.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.473440 + 1.36792i) q^{2} +(-0.458227 - 0.888835i) q^{3} +(-0.0749403 - 0.0589337i) q^{4} +(-1.20704 + 0.115258i) q^{5} +(1.43279 - 0.206005i) q^{6} +(0.607831 - 2.57498i) q^{7} +(-2.31938 + 1.49058i) q^{8} +(-0.580057 + 0.814576i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q - 4 q^{2} + 36 q^{4} + 24 q^{8} - 32 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(323\)	\(346\)	\(442\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{3}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.473440	+	1.36792i	−0.334773	+	0.967262i	0.644480	+	0.764621i	\(0.277074\pi\)
							−0.979253	+	0.202641i	\(0.935047\pi\)
\(3\)	−0.458227	−	0.888835i	−0.264557	−	0.513169i
							\(5\)	−1.20704	+	0.115258i	−0.539805	+	0.0515451i	−0.361398	−	0.932412i	\(-0.617700\pi\)
−0.178407	+	0.983957i	\(0.557094\pi\)
\(7\)	0.607831	−	2.57498i	0.229739	−	0.973252i
							\(11\)	−3.29839	+	1.14158i	−0.994502	+	0.344200i	−0.775378	−	0.631497i	\(-0.782441\pi\)
−0.219123	+	0.975697i	\(0.570320\pi\)
\(13\)	0.982048	+	3.34455i	0.272371	+	0.927611i	0.976133	+	0.217175i	\(0.0696841\pi\)
							−0.703762	+	0.710436i	\(0.748498\pi\)
\(17\)	−3.34615	−	1.33960i	−0.811562	−	0.324900i	−0.0715010	−	0.997441i	\(-0.522779\pi\)
							−0.740060	+	0.672540i	\(0.765203\pi\)
\(19\)	−5.08838	+	2.03708i	−1.16736	+	0.467339i	−0.872696	−	0.488264i	\(-0.837630\pi\)
							−0.294659	+	0.955602i	\(0.595206\pi\)
\(23\)	−1.89971	+	4.40353i	−0.396117	+	0.918200i
							\(29\)	−0.311765	−	2.16837i	−0.0578932	−	0.402656i	−0.998077	−	0.0619882i	\(-0.980256\pi\)
0.940184	−	0.340668i	\(-0.110653\pi\)
\(31\)	8.47980	−	0.403943i	1.52302	−	0.0725503i	0.730716	−	0.682681i	\(-0.239186\pi\)
							0.792301	+	0.610131i	\(0.208883\pi\)
\(37\)	−6.95571	−	4.95314i	−1.14351	−	0.814291i	−0.158599	−	0.987343i	\(-0.550698\pi\)
							−0.984913	+	0.173052i	\(0.944637\pi\)
\(41\)	−1.67461	−	0.764770i	−0.261530	−	0.119437i	0.280335	−	0.959902i	\(-0.409554\pi\)
							−0.541865	+	0.840465i	\(0.682282\pi\)
\(43\)	−6.27648	+	9.76639i	−0.957153	+	1.48936i	−0.0872210	+	0.996189i	\(0.527799\pi\)
							−0.869932	+	0.493171i	\(0.835838\pi\)
\(47\)	6.20615	−	3.58312i	0.905260	−	0.522652i	0.0263571	−	0.999653i	\(-0.491609\pi\)
							0.878903	+	0.477000i	\(0.158276\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.bf.a.10.9	✓	640
7.5	odd	6	inner	483.2.bf.a.355.24	yes	640
23.7	odd	22	inner	483.2.bf.a.283.24	yes	640
161.145	even	66	inner	483.2.bf.a.145.9	yes	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.bf.a.10.9	✓	640	1.1	even	1	trivial
483.2.bf.a.145.9	yes	640	161.145	even	66	inner
483.2.bf.a.283.24	yes	640	23.7	odd	22	inner
483.2.bf.a.355.24	yes	640	7.5	odd	6	inner