Embedded newform 483.2.bf.a.61.19

• Label: 483.2.bf.a.61.19
• Level: $483$
• Weight: $2$
• Character: 483.61
• 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			61.19
Character	\(\chi\)	\(=\)	483.61
Dual form			483.2.bf.a.388.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0678422 + 0.0533517i) q^{2} +(0.814576 + 0.580057i) q^{3} +(-0.469762 - 1.93638i) q^{4} +(1.27425 - 0.245591i) q^{5} +(0.0243156 + 0.0828113i) q^{6} +(-1.24240 - 2.33591i) q^{7} +(0.143146 - 0.313447i) q^{8} +(0.327068 + 0.945001i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q - 4 q^{2} + 36 q^{4} + 24 q^{8} - 32 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(323\)	\(346\)	\(442\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{17}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0678422	+	0.0533517i	0.0479717	+	0.0377253i	0.641857	−	0.766824i	\(-0.278164\pi\)
							−0.593885	+	0.804550i	\(0.702407\pi\)
\(3\)	0.814576	+	0.580057i	0.470296	+	0.334896i
							\(5\)	1.27425	−	0.245591i	0.569860	−	0.109831i	0.103821	−	0.994596i	\(-0.466893\pi\)
0.466038	+	0.884764i	\(0.345681\pi\)
\(7\)	−1.24240	−	2.33591i	−0.469581	−	0.882889i
							\(11\)	−2.93105	−	3.72713i	−0.883744	−	1.12377i	−0.991566	−	0.129602i	\(-0.958630\pi\)
0.107822	−	0.994170i	\(-0.465612\pi\)
\(13\)	−1.65512	−	2.57542i	−0.459049	−	0.714294i	0.532154	−	0.846648i	\(-0.321383\pi\)
							−0.991203	+	0.132354i	\(0.957747\pi\)
\(17\)	2.26441	+	2.15911i	0.549199	+	0.523660i	0.913077	−	0.407786i	\(-0.133699\pi\)
							−0.363878	+	0.931446i	\(0.618548\pi\)
\(19\)	2.09667	−	1.99917i	0.481009	−	0.458641i	−0.410357	−	0.911925i	\(-0.634596\pi\)
							0.891366	+	0.453284i	\(0.149748\pi\)
\(23\)	4.37861	+	1.95646i	0.913004	+	0.407951i
							\(29\)	−1.05739	+	0.310477i	−0.196352	+	0.0576542i	−0.378430	−	0.925630i	\(-0.623536\pi\)
0.182078	+	0.983284i	\(0.441718\pi\)
\(31\)	0.0373565	+	0.391215i	0.00670943	+	0.0702643i	0.998207	−	0.0598647i	\(-0.0190669\pi\)
							−0.991497	+	0.130129i	\(0.958461\pi\)
\(37\)	9.53414	−	3.29980i	1.56740	−	0.542484i	0.600941	−	0.799293i	\(-0.294792\pi\)
							0.966462	+	0.256809i	\(0.0826713\pi\)
\(41\)	7.20323	−	6.24163i	1.12496	−	0.974779i	0.125106	−	0.992143i	\(-0.460073\pi\)
							0.999849	+	0.0173641i	\(0.00552745\pi\)
\(43\)	−4.28446	+	1.95665i	−0.653373	+	0.298386i	−0.714386	−	0.699751i	\(-0.753294\pi\)
							0.0610132	+	0.998137i	\(0.480567\pi\)
\(47\)	0.716352	+	0.413586i	0.104491	+	0.0603277i	0.551335	−	0.834284i	\(-0.314119\pi\)
							−0.446844	+	0.894612i	\(0.647452\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.bf.a.61.19	✓	640
7.3	odd	6	inner	483.2.bf.a.199.14	yes	640
23.20	odd	22	inner	483.2.bf.a.250.14	yes	640
161.66	even	66	inner	483.2.bf.a.388.19	yes	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.bf.a.61.19	✓	640	1.1	even	1	trivial
483.2.bf.a.199.14	yes	640	7.3	odd	6	inner
483.2.bf.a.250.14	yes	640	23.20	odd	22	inner
483.2.bf.a.388.19	yes	640	161.66	even	66	inner