Embedded newform 731.2.v.a.36.8

• Label: 731.2.v.a.36.8
• Level: $731$
• Weight: $2$
• Character: 731.36
• Analytic conductor: $5.837$
• Analytic rank: $0$
• Dimension: $512$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 731 = 17 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	731.v (of order \(24\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.83706438776\)
Analytic rank:	\(0\)
Dimension:	\(512\)
Relative dimension:	\(64\) over \(\Q(\zeta_{24})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{24}]$

Embedding invariants

Embedding label			36.8
Character	\(\chi\)	\(=\)	731.36
Dual form			731.2.v.a.264.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.62148 - 1.62148i) q^{2} +(2.59315 + 1.98979i) q^{3} +3.25840i q^{4} +(2.22006 - 2.89323i) q^{5} +(-0.978330 - 7.43116i) q^{6} +(-0.233089 - 0.303768i) q^{7} +(2.04047 - 2.04047i) q^{8} +(1.98869 + 7.42189i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 512 q - 16 q^{2} - 4 q^{3} - 12 q^{5} - 4 q^{6} - 4 q^{7} + 16 q^{8} - 28 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(173\)	\(562\)
\(\chi(n)\)	\(e\left(\frac{7}{8}\right)\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.62148	−	1.62148i	−1.14656	−	1.14656i	−0.987224	−	0.159336i	\(-0.949065\pi\)
							−0.159336	−	0.987224i	\(-0.550935\pi\)
\(3\)	2.59315	+	1.98979i	1.49716	+	1.14881i	0.950198	+	0.311648i	\(0.100881\pi\)
							0.546958	+	0.837160i	\(0.315786\pi\)
\(5\)	2.22006	−	2.89323i	0.992839	−	1.29389i	0.0372142	−	0.999307i	\(-0.488152\pi\)
							0.955625	−	0.294586i	\(-0.0951817\pi\)
\(7\)	−0.233089	−	0.303768i	−0.0880994	−	0.114813i	0.747224	−	0.664572i	\(-0.231386\pi\)
							−0.835323	+	0.549759i	\(0.814720\pi\)
\(11\)	1.86522	+	0.772597i	0.562384	+	0.232947i	0.645720	−	0.763575i	\(-0.276558\pi\)
							−0.0833361	+	0.996521i	\(0.526558\pi\)
\(13\)	2.63918	+	1.52373i	0.731977	+	0.422607i	0.819145	−	0.573586i	\(-0.194448\pi\)
							−0.0871679	+	0.996194i	\(0.527782\pi\)
\(17\)	−3.21036	+	2.58719i	−0.778628	+	0.627486i
							\(19\)	1.73138	−	6.46160i	0.397206	−	1.48239i	−0.420784	−	0.907161i	\(-0.638245\pi\)
0.817990	−	0.575232i	\(-0.195088\pi\)
\(23\)	1.12836	+	8.57076i	0.235280	+	1.78713i	0.547182	+	0.837014i	\(0.315701\pi\)
							−0.311902	+	0.950114i	\(0.600966\pi\)
\(29\)	−2.31433	−	0.304687i	−0.429760	−	0.0565790i	−0.0874565	−	0.996168i	\(-0.527874\pi\)
							−0.342304	+	0.939589i	\(0.611207\pi\)
\(31\)	−6.34312	−	4.86725i	−1.13926	−	0.874184i	−0.145906	−	0.989298i	\(-0.546610\pi\)
							−0.993352	+	0.115115i	\(0.963276\pi\)
\(37\)	3.03646	+	2.32995i	0.499190	+	0.383042i	0.827445	−	0.561547i	\(-0.189794\pi\)
							−0.328254	+	0.944589i	\(0.606460\pi\)
\(41\)	0.732657	−	1.76879i	0.114422	−	0.276239i	−0.856286	−	0.516502i	\(-0.827234\pi\)
							0.970708	+	0.240263i	\(0.0772338\pi\)
\(43\)	3.88407	−	5.28337i	0.592315	−	0.805706i
							\(47\)			7.11855i			1.03835i	0.854669	+	0.519173i	\(0.173760\pi\)
−0.854669	+	0.519173i	\(0.826240\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.v.a.36.8	✓	512
17.9	even	8	inner	731.2.v.a.638.8	yes	512
43.6	even	3	inner	731.2.v.a.393.8	yes	512
731.264	even	24	inner	731.2.v.a.264.8	yes	512

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.v.a.36.8	✓	512	1.1	even	1	trivial
731.2.v.a.264.8	yes	512	731.264	even	24	inner
731.2.v.a.393.8	yes	512	43.6	even	3	inner
731.2.v.a.638.8	yes	512	17.9	even	8	inner