Embedded newform 731.2.bl.a.9.16

• Label: 731.2.bl.a.9.16
• Level: $731$
• Weight: $2$
• Character: 731.9
• Analytic conductor: $5.837$
• Analytic rank: $0$
• Dimension: $3072$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 731 = 17 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	731.bl (of order \(168\), degree \(48\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.83706438776\)
Analytic rank:	\(0\)
Dimension:	\(3072\)
Relative dimension:	\(64\) over \(\Q(\zeta_{168})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{168}]$

Embedding invariants

Embedding label			9.16
Character	\(\chi\)	\(=\)	731.9
Dual form			731.2.bl.a.325.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.69079 - 0.190506i) q^{2} +(-1.80830 - 1.08960i) q^{3} +(0.872615 + 0.199169i) q^{4} +(2.36534 - 1.54868i) q^{5} +(2.84987 + 2.18678i) q^{6} +(-2.81774 - 0.370962i) q^{7} +(1.77455 + 0.620940i) q^{8} +(0.680893 + 1.28831i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3072 q - 40 q^{2} - 52 q^{3} - 44 q^{5} - 24 q^{6} - 24 q^{7} - 72 q^{8} - 28 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(173\)	\(562\)
\(\chi(n)\)	\(e\left(\frac{1}{8}\right)\)	\(e\left(\frac{1}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.69079	−	0.190506i	−1.19557	−	0.134708i	−0.508344	−	0.861154i	\(-0.669742\pi\)
							−0.687224	+	0.726446i	\(0.741171\pi\)
\(3\)	−1.80830	−	1.08960i	−1.04402	−	0.629084i	−0.112829	−	0.993614i	\(-0.535991\pi\)
							−0.931191	+	0.364531i	\(0.881229\pi\)
\(5\)	2.36534	−	1.54868i	1.05781	−	0.692590i	0.104485	−	0.994526i	\(-0.466681\pi\)
							0.953328	+	0.301936i	\(0.0976330\pi\)
\(7\)	−2.81774	−	0.370962i	−1.06500	−	0.140210i	−0.422401	−	0.906409i	\(-0.638812\pi\)
							−0.642604	+	0.766199i	\(0.722146\pi\)
\(11\)	0.889047	+	1.25299i	0.268058	+	0.377792i	0.927413	−	0.374040i	\(-0.122028\pi\)
							−0.659355	+	0.751832i	\(0.729171\pi\)
\(13\)	4.38030	+	0.328258i	1.21488	+	0.0910425i	0.666704	−	0.745323i	\(-0.267704\pi\)
							0.548173	+	0.836365i	\(0.315324\pi\)
\(17\)	2.89940	+	2.93146i	0.703208	+	0.710984i
							\(19\)	−1.83943	+	3.48037i	−0.421994	+	0.798451i	−0.999843	−	0.0176920i	\(-0.994368\pi\)
0.577850	+	0.816143i	\(0.303892\pi\)
\(23\)	−2.20749	+	4.81205i	−0.460293	+	1.00338i	0.527778	+	0.849382i	\(0.323025\pi\)
							−0.988071	+	0.154000i	\(0.950784\pi\)
\(29\)	−0.778731	+	3.13999i	−0.144607	+	0.583081i	0.853473	+	0.521137i	\(0.174492\pi\)
							−0.998080	+	0.0619442i	\(0.980270\pi\)
\(31\)	4.13381	−	4.29138i	0.742454	−	0.770754i	−0.237197	−	0.971462i	\(-0.576229\pi\)
							0.979651	+	0.200707i	\(0.0643240\pi\)
\(37\)	1.37261	+	10.4260i	0.225656	+	1.71403i	0.611318	+	0.791385i	\(0.290640\pi\)
							−0.385662	+	0.922640i	\(0.626027\pi\)
\(41\)	3.24060	−	5.86343i	0.506097	−	0.915714i	−0.492862	−	0.870107i	\(-0.664049\pi\)
							0.998959	−	0.0456067i	\(-0.0145221\pi\)
\(43\)	−5.01566	−	4.22411i	−0.764881	−	0.644171i
							\(47\)	7.17586	+	1.63784i	1.04671	+	0.238904i	0.711095	−	0.703096i	\(-0.248200\pi\)
0.335612	+	0.942000i	\(0.391057\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.bl.a.9.16	✓	3072
17.2	even	8	inner	731.2.bl.a.138.49	yes	3072
43.24	even	21	inner	731.2.bl.a.196.49	yes	3072
731.325	even	168	inner	731.2.bl.a.325.16	yes	3072

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.bl.a.9.16	✓	3072	1.1	even	1	trivial
731.2.bl.a.138.49	yes	3072	17.2	even	8	inner
731.2.bl.a.196.49	yes	3072	43.24	even	21	inner
731.2.bl.a.325.16	yes	3072	731.325	even	168	inner