Embedded newform 731.2.u.b.52.7

• Label: 731.2.u.b.52.7
• 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			52.7
Character	\(\chi\)	\(=\)	731.52
Dual form			731.2.u.b.239.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.04859 - 1.31489i) q^{2} +(2.19207 + 0.330401i) q^{3} +(-0.184358 + 0.807723i) q^{4} +(1.73827 + 1.18513i) q^{5} +(-1.86414 - 3.22879i) q^{6} +(-0.383582 + 0.664384i) q^{7} +(-1.77514 + 0.854860i) q^{8} +(1.82928 + 0.564257i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 348 q + 6 q^{2} - 3 q^{3} - 54 q^{4} + q^{5} - 12 q^{6} + 49 q^{7} - 2 q^{8} + 8 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.04859	−	1.31489i	−0.741467	−	0.929771i	0.257870	−	0.966180i	\(-0.416980\pi\)
							−0.999337	+	0.0364090i	\(0.988408\pi\)
\(3\)	2.19207	+	0.330401i	1.26559	+	0.190757i	0.747323	−	0.664461i	\(-0.231339\pi\)
							0.518268	+	0.855218i	\(0.326577\pi\)
\(5\)	1.73827	+	1.18513i	0.777378	+	0.530008i	0.885783	−	0.464099i	\(-0.153622\pi\)
							−0.108405	+	0.994107i	\(0.534574\pi\)
\(7\)	−0.383582	+	0.664384i	−0.144980	+	0.251113i	−0.929366	−	0.369161i	\(-0.879645\pi\)
							0.784385	+	0.620274i	\(0.212979\pi\)
\(11\)	0.815638	+	3.57354i	0.245924	+	1.07746i	0.935521	+	0.353271i	\(0.114931\pi\)
							−0.689597	+	0.724193i	\(0.742212\pi\)
\(13\)	−0.500354	+	6.67677i	−0.138773	+	1.85180i	0.300915	+	0.953651i	\(0.402708\pi\)
							−0.439688	+	0.898150i	\(0.644911\pi\)
\(17\)	−0.826239	+	0.563320i	−0.200392	+	0.136625i
							\(19\)	4.36713	−	1.34708i	1.00189	−	0.309041i	0.249938	−	0.968262i	\(-0.419590\pi\)
0.751950	+	0.659221i	\(0.229114\pi\)
\(23\)	1.75964	−	1.63270i	0.366909	−	0.340442i	−0.475074	−	0.879946i	\(-0.657579\pi\)
							0.841983	+	0.539504i	\(0.181388\pi\)
\(29\)	−2.52595	+	0.380725i	−0.469057	+	0.0706989i	−0.379318	−	0.925266i	\(-0.623841\pi\)
							−0.0897387	+	0.995965i	\(0.528603\pi\)
\(31\)	1.10078	−	2.80473i	0.197705	−	0.503745i	−0.797298	−	0.603586i	\(-0.793738\pi\)
							0.995003	+	0.0998407i	\(0.0318333\pi\)
\(37\)	1.19173	+	2.06413i	0.195919	+	0.339341i	0.947201	−	0.320639i	\(-0.103898\pi\)
							−0.751283	+	0.659981i	\(0.770564\pi\)
\(41\)	7.06396	+	8.85792i	1.10321	+	1.38338i	0.916059	+	0.401043i	\(0.131352\pi\)
							0.187146	+	0.982332i	\(0.440076\pi\)
\(43\)	−3.90194	−	5.27019i	−0.595040	−	0.803696i
							\(47\)	0.547619	−	2.39927i	0.0798784	−	0.349970i	−0.919157	−	0.393892i	\(-0.871128\pi\)
0.999035	+	0.0439223i	\(0.0139854\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.u.b.52.7	✓	348
43.24	even	21	inner	731.2.u.b.239.7	yes	348

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.u.b.52.7	✓	348	1.1	even	1	trivial
731.2.u.b.239.7	yes	348	43.24	even	21	inner