Embedded newform 483.2.bf.a.19.14

• Label: 483.2.bf.a.19.14
• Level: $483$
• Weight: $2$
• Character: 483.19
• 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			19.14
Character	\(\chi\)	\(=\)	483.19
Dual form			483.2.bf.a.178.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.587934 - 0.0561409i) q^{2} +(0.690079 - 0.723734i) q^{3} +(-1.62134 - 0.312488i) q^{4} +(2.80331 - 1.44520i) q^{5} +(-0.446352 + 0.386767i) q^{6} +(-0.926191 - 2.47834i) q^{7} +(2.06907 + 0.607534i) q^{8} +(-0.0475819 - 0.998867i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q - 4 q^{2} + 36 q^{4} + 24 q^{8} - 32 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(323\)	\(346\)	\(442\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{15}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.587934	−	0.0561409i	−0.415732	−	0.0396976i	−0.114906	−	0.993376i	\(-0.536657\pi\)
							−0.300827	+	0.953679i	\(0.597263\pi\)
\(3\)	0.690079	−	0.723734i	0.398417	−	0.417848i
							\(5\)	2.80331	−	1.44520i	1.25368	−	0.646315i	0.301968	−	0.953318i	\(-0.402357\pi\)
0.951708	+	0.307003i	\(0.0993262\pi\)
\(7\)	−0.926191	−	2.47834i	−0.350067	−	0.936725i
							\(11\)	−0.0160622	−	0.168211i	−0.00484294	−	0.0507175i	0.992727	−	0.120384i	\(-0.0384126\pi\)
−0.997570	+	0.0696664i	\(0.977807\pi\)
\(13\)	−3.12346	−	0.449086i	−0.866293	−	0.124554i	−0.305181	−	0.952294i	\(-0.598717\pi\)
							−0.561112	+	0.827740i	\(0.689626\pi\)
\(17\)	0.814362	−	2.35294i	0.197512	−	0.570673i	−0.802138	−	0.597139i	\(-0.796304\pi\)
							0.999650	+	0.0264660i	\(0.00842538\pi\)
\(19\)	0.288493	+	0.833544i	0.0661847	+	0.191228i	0.973244	−	0.229775i	\(-0.0737992\pi\)
							−0.907059	+	0.421004i	\(0.861678\pi\)
\(23\)	−3.10091	+	3.65846i	−0.646585	+	0.762842i
							\(29\)	−4.98217	−	5.74973i	−0.925166	−	1.06770i	−0.997525	−	0.0703197i	\(-0.977598\pi\)
0.0723586	−	0.997379i	\(-0.476947\pi\)
\(31\)	5.54317	−	1.34476i	0.995582	−	0.241526i	0.295310	−	0.955401i	\(-0.404577\pi\)
							0.700272	+	0.713876i	\(0.253062\pi\)
\(37\)	2.43800	−	0.116136i	0.400804	−	0.0190926i	0.153788	−	0.988104i	\(-0.450853\pi\)
							0.247016	+	0.969011i	\(0.420550\pi\)
\(41\)	1.83533	−	2.85583i	0.286631	−	0.446006i	−0.667843	−	0.744302i	\(-0.732782\pi\)
							0.954473	+	0.298297i	\(0.0964185\pi\)
\(43\)	−1.82508	−	6.21564i	−0.278322	−	0.947877i	−0.973433	−	0.228974i	\(-0.926463\pi\)
							0.695111	−	0.718903i	\(-0.255355\pi\)
\(47\)	3.43570	+	1.98360i	0.501148	+	0.289338i	0.729188	−	0.684314i	\(-0.239898\pi\)
							−0.228039	+	0.973652i	\(0.573231\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.bf.a.19.14	✓	640
7.3	odd	6	inner	483.2.bf.a.157.19	yes	640
23.17	odd	22	inner	483.2.bf.a.40.19	yes	640
161.17	even	66	inner	483.2.bf.a.178.14	yes	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.bf.a.19.14	✓	640	1.1	even	1	trivial
483.2.bf.a.40.19	yes	640	23.17	odd	22	inner
483.2.bf.a.157.19	yes	640	7.3	odd	6	inner
483.2.bf.a.178.14	yes	640	161.17	even	66	inner