Embedded newform 483.2.y.b.340.16

• Label: 483.2.y.b.340.16
• Level: $483$
• Weight: $2$
• Character: 483.340
• 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			340.16
Character	\(\chi\)	\(=\)	483.340
Dual form			483.2.y.b.331.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.617230 + 2.54426i) q^{2} +(-0.327068 - 0.945001i) q^{3} +(-4.31460 + 2.22433i) q^{4} +(3.40608 - 1.36359i) q^{5} +(2.20245 - 1.41543i) q^{6} +(2.47194 - 0.943132i) q^{7} +(-4.89344 - 5.64734i) q^{8} +(-0.786053 + 0.618159i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 320 q + 2 q^{2} + 16 q^{3} + 18 q^{4} - 2 q^{5} + 18 q^{6} + 2 q^{7} - 12 q^{8} + 16 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(323\)	\(346\)	\(442\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{6}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.617230	+	2.54426i	0.436448	+	1.79906i	0.586004	+	0.810308i	\(0.300700\pi\)
							−0.149557	+	0.988753i	\(0.547785\pi\)
\(3\)	−0.327068	−	0.945001i	−0.188833	−	0.545596i
							\(5\)	3.40608	−	1.36359i	1.52325	−	0.609816i	0.549104	−	0.835754i	\(-0.314969\pi\)
0.974142	+	0.225938i	\(0.0725448\pi\)
\(7\)	2.47194	−	0.943132i	0.934307	−	0.356470i
							\(11\)	1.13440	−	4.67608i	0.342036	−	1.40989i	−0.497914	−	0.867227i	\(-0.665900\pi\)
0.839950	−	0.542664i	\(-0.182584\pi\)
\(13\)	1.90979	−	4.18186i	0.529681	−	1.15984i	−0.435962	−	0.899965i	\(-0.643592\pi\)
							0.965643	−	0.259874i	\(-0.0836809\pi\)
\(17\)	0.0881357	+	1.85020i	0.0213760	+	0.448738i	0.984030	+	0.178003i	\(0.0569637\pi\)
							−0.962654	+	0.270735i	\(0.912733\pi\)
\(19\)	−0.159166	+	3.34130i	−0.0365151	+	0.766547i	0.904151	+	0.427213i	\(0.140505\pi\)
							−0.940666	+	0.339334i	\(0.889798\pi\)
\(23\)	−3.00298	+	3.73927i	−0.626164	+	0.779692i
							\(29\)	−2.28199	+	1.46655i	−0.423755	+	0.272331i	−0.735090	−	0.677969i	\(-0.762860\pi\)
0.311335	+	0.950300i	\(0.399224\pi\)
\(31\)	−8.29125	+	1.59801i	−1.48915	+	0.287011i	−0.867955	−	0.496644i	\(-0.834566\pi\)
							−0.621197	+	0.783654i	\(0.713353\pi\)
\(37\)	−2.06147	+	1.62116i	−0.338903	+	0.266516i	−0.773076	−	0.634313i	\(-0.781283\pi\)
							0.434173	+	0.900829i	\(0.357041\pi\)
\(41\)	−1.03739	+	7.21520i	−0.162013	+	1.12682i	0.732820	+	0.680423i	\(0.238204\pi\)
							−0.894833	+	0.446402i	\(0.852705\pi\)
\(43\)	−2.60984	+	3.01192i	−0.397997	+	0.459313i	−0.919009	−	0.394236i	\(-0.871009\pi\)
							0.521012	+	0.853549i	\(0.325555\pi\)
\(47\)	0.722635	+	1.25164i	0.105407	+	0.182571i	0.913904	−	0.405929i	\(-0.133052\pi\)
							−0.808497	+	0.588500i	\(0.799719\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.y.b.340.16	yes	320
7.2	even	3	inner	483.2.y.b.478.16	yes	320
23.9	even	11	inner	483.2.y.b.193.16	✓	320
161.9	even	33	inner	483.2.y.b.331.16	yes	320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.y.b.193.16	✓	320	23.9	even	11	inner
483.2.y.b.331.16	yes	320	161.9	even	33	inner
483.2.y.b.340.16	yes	320	1.1	even	1	trivial
483.2.y.b.478.16	yes	320	7.2	even	3	inner