Embedded newform 731.2.n.a.208.8

• Label: 731.2.n.a.208.8
• Level: $731$
• Weight: $2$
• Character: 731.208
• Analytic conductor: $5.837$
• Analytic rank: $0$
• Dimension: $256$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 731 = 17 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	731.n (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.83706438776\)
Analytic rank:	\(0\)
Dimension:	\(256\)
Relative dimension:	\(64\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			208.8
Character	\(\chi\)	\(=\)	731.208
Dual form			731.2.n.a.608.57

$q$-expansion

\(f(q)\)	\(=\)	\(q-2.26939i q^{2} +(1.08136 + 0.289751i) q^{3} -3.15015 q^{4} +(1.08055 + 0.289533i) q^{5} +(0.657559 - 2.45404i) q^{6} +(-0.970179 + 0.259959i) q^{7} +2.61014i q^{8} +(-1.51268 - 0.873348i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q - 6 q^{3} - 264 q^{4} + 2 q^{5} - 2 q^{6}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/731\mathbb{Z}\right)^\times\).

\(n\)	\(173\)	\(562\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)		−	2.26939i		−	1.60470i	−0.596851	−	0.802352i	\(-0.703582\pi\)
							0.596851	−	0.802352i	\(-0.296418\pi\)
\(3\)	1.08136	+	0.289751i	0.624326	+	0.167288i	0.557094	−	0.830450i	\(-0.311916\pi\)
							0.0672324	+	0.997737i	\(0.478583\pi\)
\(5\)	1.08055	+	0.289533i	0.483237	+	0.129483i	0.492209	−	0.870477i	\(-0.336189\pi\)
							−0.00897252	+	0.999960i	\(0.502856\pi\)
\(7\)	−0.970179	+	0.259959i	−0.366693	+	0.0982551i	−0.437460	−	0.899238i	\(-0.644122\pi\)
							0.0707672	+	0.997493i	\(0.477455\pi\)
\(11\)	−3.42886	−	3.42886i	−1.03384	−	1.03384i	−0.999407	−	0.0344327i	\(-0.989038\pi\)
							−0.0344327	−	0.999407i	\(-0.510962\pi\)
\(13\)	1.79783	−	3.11392i	0.498627	−	0.863647i	−0.501372	−	0.865232i	\(-0.667171\pi\)
							0.999999	+	0.00158466i	\(0.000504412\pi\)
\(17\)	3.78503	+	1.63509i	0.918006	+	0.396567i
							\(19\)	4.38203	−	2.52997i	1.00531	−	0.580414i	0.0954928	−	0.995430i	\(-0.469557\pi\)
0.909814	+	0.415016i	\(0.136224\pi\)
\(23\)	−0.419950	+	1.56728i	−0.0875657	+	0.326800i	−0.995788	−	0.0916892i	\(-0.970773\pi\)
							0.908222	+	0.418489i	\(0.137440\pi\)
\(29\)	−1.30677	−	4.87693i	−0.242661	−	0.905624i	−0.974545	−	0.224194i	\(-0.928025\pi\)
							0.731883	−	0.681430i	\(-0.238642\pi\)
\(31\)	−1.69384	−	0.453864i	−0.304224	−	0.0815164i	0.103478	−	0.994632i	\(-0.467003\pi\)
							−0.407701	+	0.913115i	\(0.633670\pi\)
\(37\)	−0.945912	−	0.253456i	−0.155507	−	0.0416680i	0.180226	−	0.983625i	\(-0.442317\pi\)
							−0.335733	+	0.941957i	\(0.608984\pi\)
\(41\)	−3.91923	−	3.91923i	−0.612081	−	0.612081i	0.331407	−	0.943488i	\(-0.392477\pi\)
							−0.943488	+	0.331407i	\(0.892477\pi\)
\(43\)	5.53057	+	3.52317i	0.843405	+	0.537279i
							\(47\)	7.97389			1.16311			0.581556	−	0.813506i	\(-0.302444\pi\)
0.581556	+	0.813506i	\(0.302444\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.n.a.208.8	✓	256
17.13	even	4	inner	731.2.n.a.251.57	yes	256
43.6	even	3	inner	731.2.n.a.565.8	yes	256
731.608	even	12	inner	731.2.n.a.608.57	yes	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.n.a.208.8	✓	256	1.1	even	1	trivial
731.2.n.a.251.57	yes	256	17.13	even	4	inner
731.2.n.a.565.8	yes	256	43.6	even	3	inner
731.2.n.a.608.57	yes	256	731.608	even	12	inner