Embedded newform 483.2.bf.a.388.28

• Label: 483.2.bf.a.388.28
• 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			388.28
Character	\(\chi\)	\(=\)	483.388
Dual form			483.2.bf.a.61.28

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.66500 - 1.30937i) q^{2} +(-0.814576 + 0.580057i) q^{3} +(0.586254 - 2.41657i) q^{4} +(2.07093 + 0.399138i) q^{5} +(-0.596759 + 2.03237i) q^{6} +(-0.0453823 + 2.64536i) q^{7} +(-0.428227 - 0.937687i) q^{8} +(0.327068 - 0.945001i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q - 4 q^{2} + 36 q^{4} + 24 q^{8} - 32 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.66500	−	1.30937i	1.17733	−	0.925864i	0.179014	−	0.983847i	\(-0.442709\pi\)
							0.998318	+	0.0579829i	\(0.0184669\pi\)
\(3\)	−0.814576	+	0.580057i	−0.470296	+	0.334896i
							\(5\)	2.07093	+	0.399138i	0.926146	+	0.178500i	0.629958	−	0.776630i	\(-0.283072\pi\)
0.296188	+	0.955130i	\(0.404284\pi\)
\(7\)	−0.0453823	+	2.64536i	−0.0171529	+	0.999853i
							\(11\)	2.42156	−	3.07926i	0.730128	−	0.928433i	−0.269273	−	0.963064i	\(-0.586783\pi\)
0.999400	+	0.0346313i	\(0.0110257\pi\)
\(13\)	−0.763770	+	1.18845i	−0.211832	+	0.329617i	−0.930868	−	0.365356i	\(-0.880947\pi\)
							0.719036	+	0.694973i	\(0.244584\pi\)
\(17\)	−0.200736	+	0.191401i	−0.0486856	+	0.0464216i	−0.714031	−	0.700114i	\(-0.753133\pi\)
							0.665346	+	0.746535i	\(0.268284\pi\)
\(19\)	0.417351	+	0.397943i	0.0957468	+	0.0912944i	0.736441	−	0.676502i	\(-0.236505\pi\)
							−0.640694	+	0.767796i	\(0.721353\pi\)
\(23\)	2.38671	−	4.15976i	0.497663	−	0.867370i
							\(29\)	−5.20891	−	1.52947i	−0.967270	−	0.284016i	−0.240310	−	0.970696i	\(-0.577249\pi\)
−0.726960	+	0.686680i	\(0.759067\pi\)
\(31\)	−0.233410	+	2.44438i	−0.0419217	+	0.439024i	0.950007	+	0.312228i	\(0.101075\pi\)
							−0.991929	+	0.126796i	\(0.959531\pi\)
\(37\)	−3.04978	−	1.05554i	−0.501380	−	0.173529i	0.0646594	−	0.997907i	\(-0.479404\pi\)
							−0.566040	+	0.824378i	\(0.691525\pi\)
\(41\)	−4.70721	−	4.07882i	−0.735142	−	0.637004i	0.204617	−	0.978842i	\(-0.434405\pi\)
							−0.939759	+	0.341838i	\(0.888951\pi\)
\(43\)	−3.42515	−	1.56421i	−0.522330	−	0.238540i	0.136760	−	0.990604i	\(-0.456331\pi\)
							−0.659090	+	0.752064i	\(0.729058\pi\)
\(47\)	1.68269	−	0.971502i	0.245446	−	0.141708i	−0.372231	−	0.928140i	\(-0.621407\pi\)
							0.617677	+	0.786432i	\(0.288074\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.bf.a.388.28	yes	640
7.5	odd	6	inner	483.2.bf.a.250.5	yes	640
23.15	odd	22	inner	483.2.bf.a.199.5	yes	640
161.61	even	66	inner	483.2.bf.a.61.28	✓	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.bf.a.61.28	✓	640	161.61	even	66	inner
483.2.bf.a.199.5	yes	640	23.15	odd	22	inner
483.2.bf.a.250.5	yes	640	7.5	odd	6	inner
483.2.bf.a.388.28	yes	640	1.1	even	1	trivial