Embedded newform 731.2.u.b.52.16

• Label: 731.2.u.b.52.16
• 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			52.16
Character	\(\chi\)	\(=\)	731.52
Dual form			731.2.u.b.239.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.176668 + 0.221534i) q^{2} +(3.18926 + 0.480704i) q^{3} +(0.427176 - 1.87158i) q^{4} +(-2.99852 - 2.04436i) q^{5} +(0.456948 + 0.791457i) q^{6} +(0.233016 - 0.403596i) q^{7} +(1.00067 - 0.481898i) q^{8} +(7.07361 + 2.18192i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 348 q + 6 q^{2} - 3 q^{3} - 54 q^{4} + q^{5} - 12 q^{6} + 49 q^{7} - 2 q^{8} + 8 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.176668	+	0.221534i	0.124923	+	0.156648i	0.840360	−	0.542029i	\(-0.182344\pi\)
							−0.715437	+	0.698677i	\(0.753772\pi\)
\(3\)	3.18926	+	0.480704i	1.84132	+	0.277535i	0.975042	−	0.222022i	\(-0.0712657\pi\)
							0.866281	+	0.499557i	\(0.166504\pi\)
\(5\)	−2.99852	−	2.04436i	−1.34098	−	0.914263i	−0.341308	−	0.939951i	\(-0.610870\pi\)
							−0.999670	+	0.0256880i	\(0.991822\pi\)
\(7\)	0.233016	−	0.403596i	0.0880719	−	0.152545i	−0.818624	−	0.574330i	\(-0.805263\pi\)
							0.906696	+	0.421785i	\(0.138596\pi\)
\(11\)	−0.453632	−	1.98749i	−0.136775	−	0.599251i	−0.996132	−	0.0878738i	\(-0.971993\pi\)
							0.859357	−	0.511377i	\(-0.170864\pi\)
\(13\)	0.00519579	−	0.0693330i	0.00144105	−	0.0192295i	−0.996437	−	0.0843414i	\(-0.973121\pi\)
							0.997878	+	0.0651119i	\(0.0207404\pi\)
\(17\)	−0.826239	+	0.563320i	−0.200392	+	0.136625i
							\(19\)	2.20181	−	0.679170i	0.505131	−	0.155812i	−0.0317078	−	0.999497i	\(-0.510095\pi\)
0.536839	+	0.843685i	\(0.319618\pi\)
\(23\)	−3.95609	+	3.67071i	−0.824902	+	0.765397i	−0.974747	−	0.223313i	\(-0.928313\pi\)
							0.149845	+	0.988709i	\(0.452122\pi\)
\(29\)	−1.49925	+	0.225975i	−0.278403	+	0.0419625i	−0.286760	−	0.958003i	\(-0.592578\pi\)
							0.00835657	+	0.999965i	\(0.497340\pi\)
\(31\)	2.64563	−	6.74096i	0.475169	−	1.21071i	−0.469572	−	0.882894i	\(-0.655592\pi\)
							0.944741	−	0.327817i	\(-0.106313\pi\)
\(37\)	1.40914	+	2.44071i	0.231661	+	0.401249i	0.958297	−	0.285774i	\(-0.0922505\pi\)
							−0.726636	+	0.687023i	\(0.758917\pi\)
\(41\)	7.16739	+	8.98762i	1.11936	+	1.40363i	0.904231	+	0.427044i	\(0.140445\pi\)
							0.215127	+	0.976586i	\(0.430983\pi\)
\(43\)	2.20961	−	6.17395i	0.336962	−	0.941518i
							\(47\)	1.95318	−	8.55742i	0.284900	−	1.24823i	−0.606527	−	0.795063i	\(-0.707438\pi\)
0.891427	−	0.453165i	\(-0.149705\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.u.b.52.16	✓	348
43.24	even	21	inner	731.2.u.b.239.16	yes	348

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.u.b.52.16	✓	348	1.1	even	1	trivial
731.2.u.b.239.16	yes	348	43.24	even	21	inner