Embedded newform 483.2.bf.a.388.5

• Label: 483.2.bf.a.388.5
• 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			388.5
Character	\(\chi\)	\(=\)	483.388
Dual form			483.2.bf.a.61.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.85581 + 1.45943i) q^{2} +(-0.814576 + 0.580057i) q^{3} +(0.842596 - 3.47323i) q^{4} +(-0.200639 - 0.0386700i) q^{5} +(0.665150 - 2.26529i) q^{6} +(-0.836957 + 2.50988i) q^{7} +(1.54370 + 3.38022i) q^{8} +(0.327068 - 0.945001i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q - 4 q^{2} + 36 q^{4} + 24 q^{8} - 32 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.85581	+	1.45943i	−1.31226	+	1.03197i	−0.315719	+	0.948853i	\(0.602246\pi\)
							−0.996540	+	0.0831185i	\(0.973512\pi\)
\(3\)	−0.814576	+	0.580057i	−0.470296	+	0.334896i
							\(5\)	−0.200639	−	0.0386700i	−0.0897284	−	0.0172937i	0.144189	−	0.989550i	\(-0.453943\pi\)
−0.233918	+	0.972256i	\(0.575155\pi\)
\(7\)	−0.836957	+	2.50988i	−0.316340	+	0.948646i
							\(11\)	−2.29834	+	2.92258i	−0.692976	+	0.881191i	−0.997453	−	0.0713289i	\(-0.977276\pi\)
0.304477	+	0.952520i	\(0.401518\pi\)
\(13\)	1.47573	−	2.29628i	0.409294	−	0.636874i	−0.574009	−	0.818849i	\(-0.694613\pi\)
							0.983303	+	0.181975i	\(0.0582489\pi\)
\(17\)	−4.66895	+	4.45183i	−1.13239	+	1.07973i	−0.136360	+	0.990659i	\(0.543540\pi\)
							−0.996026	+	0.0890683i	\(0.971611\pi\)
\(19\)	−1.43209	−	1.36550i	−0.328544	−	0.313266i	0.507796	−	0.861478i	\(-0.330461\pi\)
							−0.836340	+	0.548211i	\(0.815309\pi\)
\(23\)	3.55679	−	3.21702i	0.741642	−	0.670796i
							\(29\)	−7.84394	−	2.30319i	−1.45658	−	0.427692i	−0.544871	−	0.838520i	\(-0.683421\pi\)
−0.911713	+	0.410828i	\(0.865240\pi\)
\(31\)	0.885823	−	9.27676i	0.159099	−	1.66616i	−0.465378	−	0.885112i	\(-0.654082\pi\)
							0.624477	−	0.781043i	\(-0.285312\pi\)
\(37\)	0.383460	+	0.132717i	0.0630405	+	0.0218185i	0.358406	−	0.933566i	\(-0.383320\pi\)
							−0.295366	+	0.955384i	\(0.595442\pi\)
\(41\)	1.23179	+	1.06736i	0.192374	+	0.166693i	0.745722	−	0.666257i	\(-0.232105\pi\)
							−0.553348	+	0.832950i	\(0.686650\pi\)
\(43\)	9.22993	+	4.21517i	1.40755	+	0.642807i	0.966966	−	0.254904i	\(-0.0820440\pi\)
							0.440584	+	0.897711i	\(0.354771\pi\)
\(47\)	11.1476	−	6.43608i	1.62605	−	0.938799i	0.640791	−	0.767715i	\(-0.278606\pi\)
							0.985257	−	0.171084i	\(-0.0547268\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.bf.a.388.5	yes	640
7.5	odd	6	inner	483.2.bf.a.250.28	yes	640
23.15	odd	22	inner	483.2.bf.a.199.28	yes	640
161.61	even	66	inner	483.2.bf.a.61.5	✓	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.bf.a.61.5	✓	640	161.61	even	66	inner
483.2.bf.a.199.28	yes	640	23.15	odd	22	inner
483.2.bf.a.250.28	yes	640	7.5	odd	6	inner
483.2.bf.a.388.5	yes	640	1.1	even	1	trivial