Embedded newform 483.2.bf.a.388.1

• Label: 483.2.bf.a.388.1
• 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			388.1
Character	\(\chi\)	\(=\)	483.388
Dual form			483.2.bf.a.61.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.16982 + 1.70637i) q^{2} +(-0.814576 + 0.580057i) q^{3} +(1.32492 - 5.46141i) q^{4} +(-1.29442 - 0.249478i) q^{5} +(0.777695 - 2.64859i) q^{6} +(1.94687 - 1.79157i) q^{7} +(4.15089 + 9.08919i) q^{8} +(0.327068 - 0.945001i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q - 4 q^{2} + 36 q^{4} + 24 q^{8} - 32 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.16982	+	1.70637i	−1.53430	+	1.20658i	−0.632392	+	0.774649i	\(0.717927\pi\)
							−0.901905	+	0.431935i	\(0.857831\pi\)
\(3\)	−0.814576	+	0.580057i	−0.470296	+	0.334896i
							\(5\)	−1.29442	−	0.249478i	−0.578881	−	0.111570i	−0.108598	−	0.994086i	\(-0.534636\pi\)
−0.470283	+	0.882516i	\(0.655848\pi\)
\(7\)	1.94687	−	1.79157i	0.735847	−	0.677148i
							\(11\)	−0.534770	+	0.680015i	−0.161239	+	0.205032i	−0.859959	−	0.510363i	\(-0.829511\pi\)
0.698720	+	0.715395i	\(0.253753\pi\)
\(13\)	−2.20144	+	3.42550i	−0.610569	+	0.950064i	0.389015	+	0.921231i	\(0.372815\pi\)
							−0.999584	+	0.0288324i	\(0.990821\pi\)
\(17\)	2.08545	−	1.98847i	0.505796	−	0.482276i	−0.393679	−	0.919248i	\(-0.628798\pi\)
							0.899475	+	0.436972i	\(0.143949\pi\)
\(19\)	−0.858637	−	0.818709i	−0.196985	−	0.187825i	0.585166	−	0.810913i	\(-0.301029\pi\)
							−0.782151	+	0.623089i	\(0.785878\pi\)
\(23\)	−4.16865	+	2.37115i	−0.869223	+	0.494419i
							\(29\)	−5.72974	−	1.68240i	−1.06399	−	0.312414i	−0.297531	−	0.954712i	\(-0.596163\pi\)
−0.766455	+	0.642298i	\(0.777981\pi\)
\(31\)	−0.387507	+	4.05815i	−0.0695982	+	0.728866i	0.891566	+	0.452890i	\(0.149607\pi\)
							−0.961165	+	0.275976i	\(0.910999\pi\)
\(37\)	6.47365	+	2.24055i	1.06426	+	0.368344i	0.802416	−	0.596765i	\(-0.203548\pi\)
							0.261846	+	0.965110i	\(0.415669\pi\)
\(41\)	−4.75657	−	4.12159i	−0.742851	−	0.643684i	0.198890	−	0.980022i	\(-0.436266\pi\)
							−0.941742	+	0.336337i	\(0.890812\pi\)
\(43\)	−2.05450	−	0.938259i	−0.313308	−	0.143083i	0.252554	−	0.967583i	\(-0.418729\pi\)
							−0.565862	+	0.824500i	\(0.691457\pi\)
\(47\)	−8.33063	+	4.80969i	−1.21515	+	0.701565i	−0.963876	−	0.266351i	\(-0.914182\pi\)
							−0.251271	+	0.967917i	\(0.580849\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.bf.a.388.1	yes	640
7.5	odd	6	inner	483.2.bf.a.250.32	yes	640
23.15	odd	22	inner	483.2.bf.a.199.32	yes	640
161.61	even	66	inner	483.2.bf.a.61.1	✓	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.bf.a.61.1	✓	640	161.61	even	66	inner
483.2.bf.a.199.32	yes	640	23.15	odd	22	inner
483.2.bf.a.250.32	yes	640	7.5	odd	6	inner
483.2.bf.a.388.1	yes	640	1.1	even	1	trivial