Embedded newform 483.2.bf.a.388.23

• Label: 483.2.bf.a.388.23
• Level: $483$
• Weight: $2$
• Character: 483.388
• 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			388.23
Character	\(\chi\)	\(=\)	483.388
Dual form			483.2.bf.a.61.23

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.04132 - 0.818906i) q^{2} +(0.814576 - 0.580057i) q^{3} +(-0.0577704 + 0.238133i) q^{4} +(-1.14488 - 0.220658i) q^{5} +(0.373225 - 1.27109i) q^{6} +(1.26894 + 2.32159i) q^{7} +(1.23549 + 2.70535i) q^{8} +(0.327068 - 0.945001i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q - 4 q^{2} + 36 q^{4} + 24 q^{8} - 32 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.04132	−	0.818906i	0.736327	−	0.579054i	−0.178000	−	0.984030i	\(-0.556963\pi\)
							0.914327	+	0.404977i	\(0.132720\pi\)
\(3\)	0.814576	−	0.580057i	0.470296	−	0.334896i
							\(5\)	−1.14488	−	0.220658i	−0.512007	−	0.0986813i	−0.0732973	−	0.997310i	\(-0.523352\pi\)
−0.438710	+	0.898629i	\(0.644564\pi\)
\(7\)	1.26894	+	2.32159i	0.479614	+	0.877480i
							\(11\)	1.25703	−	1.59844i	0.379008	−	0.481948i	−0.558701	−	0.829369i	\(-0.688700\pi\)
0.937709	+	0.347421i	\(0.112943\pi\)
\(13\)	3.57332	−	5.56019i	0.991060	−	1.54212i	0.159117	−	0.987260i	\(-0.449135\pi\)
							0.831944	−	0.554860i	\(-0.187228\pi\)
\(17\)	2.95016	−	2.81298i	0.715520	−	0.682247i	−0.242110	−	0.970249i	\(-0.577839\pi\)
							0.957630	+	0.288002i	\(0.0929909\pi\)
\(19\)	5.18103	+	4.94010i	1.18861	+	1.13334i	0.988127	+	0.153640i	\(0.0490997\pi\)
							0.200483	+	0.979697i	\(0.435749\pi\)
\(23\)	−3.94228	+	2.73101i	−0.822023	+	0.569454i
							\(29\)	−4.96774	−	1.45866i	−0.922487	−	0.270867i	−0.214199	−	0.976790i	\(-0.568714\pi\)
−0.708288	+	0.705923i	\(0.750532\pi\)
\(31\)	−0.777700	+	8.14444i	−0.139679	+	1.46278i	0.605460	+	0.795876i	\(0.292989\pi\)
							−0.745139	+	0.666909i	\(0.767617\pi\)
\(37\)	−1.43231	−	0.495728i	−0.235471	−	0.0814972i	0.206784	−	0.978387i	\(-0.433700\pi\)
							−0.442255	+	0.896889i	\(0.645821\pi\)
\(41\)	−5.03014	−	4.35865i	−0.785577	−	0.680706i	0.166678	−	0.986011i	\(-0.446696\pi\)
							−0.952254	+	0.305305i	\(0.901241\pi\)
\(43\)	−2.80228	−	1.27976i	−0.427343	−	0.195161i	0.190120	−	0.981761i	\(-0.439112\pi\)
							−0.617463	+	0.786600i	\(0.711840\pi\)
\(47\)	−5.85817	+	3.38222i	−0.854503	+	0.493347i	−0.862167	−	0.506623i	\(-0.830893\pi\)
							0.00766493	+	0.999971i	\(0.497560\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.bf.a.388.23	yes	640
7.5	odd	6	inner	483.2.bf.a.250.10	yes	640
23.15	odd	22	inner	483.2.bf.a.199.10	yes	640
161.61	even	66	inner	483.2.bf.a.61.23	✓	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.bf.a.61.23	✓	640	161.61	even	66	inner
483.2.bf.a.199.10	yes	640	23.15	odd	22	inner
483.2.bf.a.250.10	yes	640	7.5	odd	6	inner
483.2.bf.a.388.23	yes	640	1.1	even	1	trivial