Embedded newform 731.2.u.b.52.6

• Label: 731.2.u.b.52.6
• Level: $731$
• Weight: $2$
• Character: 731.52
• Analytic conductor: $5.837$
• Analytic rank: $0$
• Dimension: $348$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 731 = 17 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	731.u (of order \(21\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.83706438776\)
Analytic rank:	\(0\)
Dimension:	\(348\)
Relative dimension:	\(29\) over \(\Q(\zeta_{21})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{21}]$

Embedding invariants

Embedding label			52.6
Character	\(\chi\)	\(=\)	731.52
Dual form			731.2.u.b.239.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.15937 - 1.45380i) q^{2} +(-0.101371 - 0.0152793i) q^{3} +(-0.324361 + 1.42112i) q^{4} +(-3.42704 - 2.33652i) q^{5} +(0.0953136 + 0.165088i) q^{6} +(0.698039 - 1.20904i) q^{7} +(-0.908594 + 0.437556i) q^{8} +(-2.85668 - 0.881168i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 348 q + 6 q^{2} - 3 q^{3} - 54 q^{4} + q^{5} - 12 q^{6} + 49 q^{7} - 2 q^{8} + 8 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(173\)	\(562\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.15937	−	1.45380i	−0.819796	−	1.02799i	−0.999023	−	0.0441839i	\(-0.985931\pi\)
							0.179228	−	0.983808i	\(-0.442640\pi\)
\(3\)	−0.101371	−	0.0152793i	−0.0585268	−	0.00882150i	0.119714	−	0.992808i	\(-0.461802\pi\)
							−0.178240	+	0.983987i	\(0.557040\pi\)
\(5\)	−3.42704	−	2.33652i	−1.53262	−	1.04492i	−0.976937	−	0.213527i	\(-0.931505\pi\)
							−0.555682	−	0.831395i	\(-0.687543\pi\)
\(7\)	0.698039	−	1.20904i	0.263834	−	0.456973i	−0.703424	−	0.710771i	\(-0.748346\pi\)
							0.967257	+	0.253797i	\(0.0816796\pi\)
\(11\)	−0.570213	−	2.49826i	−0.171926	−	0.753255i	−0.985205	−	0.171381i	\(-0.945177\pi\)
							0.813279	−	0.581874i	\(-0.197680\pi\)
\(13\)	0.512185	−	6.83463i	0.142055	−	1.89559i	−0.238978	−	0.971025i	\(-0.576812\pi\)
							0.381033	−	0.924562i	\(-0.375569\pi\)
\(17\)	−0.826239	+	0.563320i	−0.200392	+	0.136625i
							\(19\)	0.748112	−	0.230762i	0.171629	−	0.0529404i	−0.207750	−	0.978182i	\(-0.566614\pi\)
0.379378	+	0.925242i	\(0.376138\pi\)
\(23\)	−1.87414	+	1.73895i	−0.390786	+	0.362596i	−0.850958	−	0.525233i	\(-0.823978\pi\)
							0.460172	+	0.887830i	\(0.347788\pi\)
\(29\)	3.86132	−	0.582000i	0.717028	−	0.108075i	0.219613	−	0.975587i	\(-0.429521\pi\)
							0.497415	+	0.867512i	\(0.334282\pi\)
\(31\)	0.483482	−	1.23189i	0.0868359	−	0.221254i	−0.880859	−	0.473379i	\(-0.843034\pi\)
							0.967695	+	0.252125i	\(0.0811293\pi\)
\(37\)	−1.42024	−	2.45992i	−0.233485	−	0.404408i	0.725346	−	0.688384i	\(-0.241680\pi\)
							−0.958831	+	0.283976i	\(0.908346\pi\)
\(41\)	2.74620	+	3.44363i	0.428885	+	0.537804i	0.948576	−	0.316550i	\(-0.102524\pi\)
							−0.519691	+	0.854354i	\(0.673953\pi\)
\(43\)	−6.38357	+	1.50000i	−0.973486	+	0.228748i
							\(47\)	−0.327690	+	1.43570i	−0.0477985	+	0.209419i	−0.993188	−	0.116524i	\(-0.962825\pi\)
0.945389	+	0.325943i	\(0.105682\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.u.b.52.6	✓	348
43.24	even	21	inner	731.2.u.b.239.6	yes	348

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.u.b.52.6	✓	348	1.1	even	1	trivial
731.2.u.b.239.6	yes	348	43.24	even	21	inner