Embedded newform 483.2.bf.a.10.29

• Label: 483.2.bf.a.10.29
• Level: $483$
• Weight: $2$
• Character: 483.10
• 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			10.29
Character	\(\chi\)	\(=\)	483.10
Dual form			483.2.bf.a.145.29

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.761915 - 2.20141i) q^{2} +(0.458227 + 0.888835i) q^{3} +(-2.69358 - 2.11825i) q^{4} +(0.293147 - 0.0279922i) q^{5} +(2.30582 - 0.331527i) q^{6} +(2.64245 + 0.132130i) q^{7} +(-2.79597 + 1.79686i) q^{8} +(-0.580057 + 0.814576i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q - 4 q^{2} + 36 q^{4} + 24 q^{8} - 32 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(323\)	\(346\)	\(442\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{3}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.761915	−	2.20141i	0.538755	−	1.55663i	−0.266581	−	0.963812i	\(-0.585894\pi\)
							0.805336	−	0.592818i	\(-0.201985\pi\)
\(3\)	0.458227	+	0.888835i	0.264557	+	0.513169i
							\(5\)	0.293147	−	0.0279922i	0.131099	−	0.0125185i	−0.0293000	−	0.999571i	\(-0.509328\pi\)
0.160399	+	0.987052i	\(0.448722\pi\)
\(7\)	2.64245	+	0.132130i	0.998752	+	0.0499404i
							\(11\)	3.62298	−	1.25392i	1.09237	−	0.378072i	0.279278	−	0.960210i	\(-0.409905\pi\)
0.813090	+	0.582138i	\(0.197784\pi\)
\(13\)	0.269748	+	0.918678i	0.0748147	+	0.254796i	0.988405	−	0.151840i	\(-0.0485200\pi\)
							−0.913590	+	0.406636i	\(0.866702\pi\)
\(17\)	−5.89142	−	2.35857i	−1.42888	−	0.572037i	−0.477108	−	0.878844i	\(-0.658315\pi\)
							−0.951772	+	0.306807i	\(0.900739\pi\)
\(19\)	6.87516	−	2.75240i	1.57727	−	0.631444i	0.592542	−	0.805540i	\(-0.298125\pi\)
							0.984729	+	0.174096i	\(0.0557003\pi\)
\(23\)	−4.65407	+	1.15742i	−0.970441	+	0.241340i
							\(29\)	0.806382	+	5.60851i	0.149741	+	1.04147i	0.916642	+	0.399708i	\(0.130889\pi\)
−0.766901	+	0.641765i	\(0.778202\pi\)
\(31\)	7.80134	−	0.371623i	1.40116	−	0.0667455i	0.666665	−	0.745358i	\(-0.267721\pi\)
							0.734497	+	0.678612i	\(0.237418\pi\)
\(37\)	−5.92195	−	4.21700i	−0.973563	−	0.693271i	−0.0216901	−	0.999765i	\(-0.506905\pi\)
							−0.951873	+	0.306494i	\(0.900844\pi\)
\(41\)	1.00238	+	0.457770i	0.156545	+	0.0714917i	0.492147	−	0.870512i	\(-0.336212\pi\)
							−0.335602	+	0.942004i	\(0.608940\pi\)
\(43\)	−4.84633	+	7.54104i	−0.739059	+	1.15000i	0.244542	+	0.969639i	\(0.421362\pi\)
							−0.983601	+	0.180359i	\(0.942274\pi\)
\(47\)	4.82632	−	2.78648i	0.703991	−	0.406450i	−0.104841	−	0.994489i	\(-0.533433\pi\)
							0.808832	+	0.588039i	\(0.200100\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	483.2.bf.a.10.29	✓	640
7.5	odd	6	inner	483.2.bf.a.355.4	yes	640
23.7	odd	22	inner	483.2.bf.a.283.4	yes	640
161.145	even	66	inner	483.2.bf.a.145.29	yes	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.bf.a.10.29	✓	640	1.1	even	1	trivial
483.2.bf.a.145.29	yes	640	161.145	even	66	inner
483.2.bf.a.283.4	yes	640	23.7	odd	22	inner
483.2.bf.a.355.4	yes	640	7.5	odd	6	inner