Embedded newform 731.2.v.a.36.6

• Label: 731.2.v.a.36.6
• 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			36.6
Character	\(\chi\)	\(=\)	731.36
Dual form			731.2.v.a.264.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.65128 - 1.65128i) q^{2} +(1.93435 + 1.48428i) q^{3} +3.45346i q^{4} +(-0.809599 + 1.05509i) q^{5} +(-0.743194 - 5.64512i) q^{6} +(-1.90304 - 2.48008i) q^{7} +(2.40007 - 2.40007i) q^{8} +(0.762170 + 2.84446i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 512 q - 16 q^{2} - 4 q^{3} - 12 q^{5} - 4 q^{6} - 4 q^{7} + 16 q^{8} - 28 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.65128	−	1.65128i	−1.16763	−	1.16763i	−0.982763	−	0.184869i	\(-0.940814\pi\)
							−0.184869	−	0.982763i	\(-0.559186\pi\)
\(3\)	1.93435	+	1.48428i	1.11680	+	0.856949i	0.990834	−	0.135086i	\(-0.0431309\pi\)
							0.125964	+	0.992035i	\(0.459798\pi\)
\(5\)	−0.809599	+	1.05509i	−0.362064	+	0.471851i	−0.938494	−	0.345296i	\(-0.887779\pi\)
							0.576430	+	0.817146i	\(0.304445\pi\)
\(7\)	−1.90304	−	2.48008i	−0.719280	−	0.937384i	0.280467	−	0.959864i	\(-0.409511\pi\)
							−0.999747	+	0.0224796i	\(0.992844\pi\)
\(11\)	2.68526	+	1.11227i	0.809636	+	0.335362i	0.748809	−	0.662786i	\(-0.230626\pi\)
							0.0608270	+	0.998148i	\(0.480626\pi\)
\(13\)	0.352891	+	0.203742i	0.0978744	+	0.0565078i	0.548138	−	0.836388i	\(-0.315337\pi\)
							−0.450264	+	0.892896i	\(0.648670\pi\)
\(17\)	4.12226	+	0.0834364i	0.999795	+	0.0202363i
							\(19\)	−1.89059	+	7.05578i	−0.433731	+	1.61871i	0.310353	+	0.950621i	\(0.399553\pi\)
−0.744084	+	0.668086i	\(0.767114\pi\)
\(23\)	−0.132059	−	1.00309i	−0.0275363	−	0.209159i	0.972146	−	0.234377i	\(-0.0753052\pi\)
							−0.999682	+	0.0252186i	\(0.991972\pi\)
\(29\)	−7.37041	−	0.970333i	−1.36865	−	0.180186i	−0.589878	−	0.807492i	\(-0.700824\pi\)
							−0.778773	+	0.627306i	\(0.784158\pi\)
\(31\)	2.18617	+	1.67751i	0.392649	+	0.301290i	0.786152	−	0.618033i	\(-0.212070\pi\)
							−0.393503	+	0.919323i	\(0.628737\pi\)
\(37\)	6.46616	+	4.96166i	1.06303	+	0.815691i	0.983458	−	0.181138i	\(-0.0579780\pi\)
							0.0795719	+	0.996829i	\(0.474645\pi\)
\(41\)	−2.04172	+	4.92914i	−0.318863	+	0.769802i	0.680452	+	0.732792i	\(0.261783\pi\)
							−0.999315	+	0.0370100i	\(0.988217\pi\)
\(43\)	5.82422	+	3.01304i	0.888186	+	0.459485i
							\(47\)			10.0114i			1.46031i	0.683281	+	0.730156i	\(0.260552\pi\)
−0.683281	+	0.730156i	\(0.739448\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.v.a.36.6	✓	512
17.9	even	8	inner	731.2.v.a.638.6	yes	512
43.6	even	3	inner	731.2.v.a.393.6	yes	512
731.264	even	24	inner	731.2.v.a.264.6	yes	512

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.v.a.36.6	✓	512	1.1	even	1	trivial
731.2.v.a.264.6	yes	512	731.264	even	24	inner
731.2.v.a.393.6	yes	512	43.6	even	3	inner
731.2.v.a.638.6	yes	512	17.9	even	8	inner