Embedded newform 731.2.p.a.16.7

• Label: 731.2.p.a.16.7
• Level: $731$
• Weight: $2$
• Character: 731.16
• Analytic conductor: $5.837$
• Analytic rank: $0$
• Dimension: $384$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 731 = 17 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	731.p (of order \(14\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.83706438776\)
Analytic rank:	\(0\)
Dimension:	\(384\)
Relative dimension:	\(64\) over \(\Q(\zeta_{14})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{14}]$

Embedding invariants

Embedding label			16.7
Character	\(\chi\)	\(=\)	731.16
Dual form			731.2.p.a.594.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.521303 - 2.28398i) q^{2} +(-1.00960 - 0.230435i) q^{3} +(-3.14286 + 1.51352i) q^{4} +(-2.97746 + 2.37445i) q^{5} +2.42604i q^{6} +3.68726i q^{7} +(2.17391 + 2.72600i) q^{8} +(-1.73671 - 0.836354i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 384 q - 12 q^{2} - 72 q^{4} - 8 q^{8} + 62 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(173\)	\(562\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{4}{7}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.521303	−	2.28398i	−0.368617	−	1.61502i	−0.730583	−	0.682824i	\(-0.760751\pi\)
							0.361966	−	0.932191i	\(-0.382106\pi\)
\(3\)	−1.00960	−	0.230435i	−0.582895	−	0.133042i	−0.0791021	−	0.996867i	\(-0.525205\pi\)
							−0.503793	+	0.863825i	\(0.668062\pi\)
\(5\)	−2.97746	+	2.37445i	−1.33156	+	1.06188i	−0.338917	+	0.940816i	\(0.610061\pi\)
							−0.992644	+	0.121068i	\(0.961368\pi\)
\(7\)			3.68726i			1.39365i	0.717239	+	0.696827i	\(0.245405\pi\)
							−0.717239	+	0.696827i	\(0.754595\pi\)
\(11\)	1.17682	−	2.44368i	0.354823	−	0.736798i	−0.644797	−	0.764354i	\(-0.723058\pi\)
							0.999620	+	0.0275557i	\(0.00877237\pi\)
\(13\)	0.443006	+	0.555512i	0.122868	+	0.154071i	0.839461	−	0.543420i	\(-0.182871\pi\)
							−0.716593	+	0.697491i	\(0.754299\pi\)
\(17\)	0.494166	−	4.09338i	0.119853	−	0.992792i
							\(19\)	−1.54294	+	0.743041i	−0.353975	+	0.170465i	−0.602417	−	0.798181i	\(-0.705796\pi\)
0.248442	+	0.968647i	\(0.420081\pi\)
\(23\)	0.363947	−	0.755744i	0.0758882	−	0.157583i	−0.859573	−	0.511013i	\(-0.829270\pi\)
							0.935461	+	0.353430i	\(0.114985\pi\)
\(29\)	2.76515	−	0.631127i	0.513475	−	0.117197i	0.0420709	−	0.999115i	\(-0.486604\pi\)
							0.471404	+	0.881917i	\(0.343747\pi\)
\(31\)	−9.01589	+	2.05782i	−1.61930	+	0.369595i	−0.933608	−	0.358296i	\(-0.883358\pi\)
							−0.685693	+	0.727891i	\(0.740501\pi\)
\(37\)		−	6.28732i		−	1.03363i	−0.856097	−	0.516815i	\(-0.827118\pi\)
							0.856097	−	0.516815i	\(-0.172882\pi\)
\(41\)	8.33376	−	1.90213i	1.30152	−	0.297062i	0.485065	−	0.874478i	\(-0.338796\pi\)
							0.816450	+	0.577416i	\(0.195939\pi\)
\(43\)	6.34017	−	1.67399i	0.966867	−	0.255281i
							\(47\)	5.42101	−	2.61062i	0.790736	−	0.380798i	0.00549205	−	0.999985i	\(-0.498252\pi\)
0.785244	+	0.619187i	\(0.212538\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.p.a.16.7	✓	384
17.16	even	2	inner	731.2.p.a.16.8	yes	384
43.35	even	7	inner	731.2.p.a.594.8	yes	384
731.594	even	14	inner	731.2.p.a.594.7	yes	384

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.p.a.16.7	✓	384	1.1	even	1	trivial
731.2.p.a.16.8	yes	384	17.16	even	2	inner
731.2.p.a.594.7	yes	384	731.594	even	14	inner
731.2.p.a.594.8	yes	384	43.35	even	7	inner