Embedded newform 731.2.v.a.36.7

• Label: 731.2.v.a.36.7
• 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			36.7
Character	\(\chi\)	\(=\)	731.36
Dual form			731.2.v.a.264.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.62967 - 1.62967i) q^{2} +(0.979757 + 0.751794i) q^{3} +3.31164i q^{4} +(1.53210 - 1.99667i) q^{5} +(-0.371504 - 2.82185i) q^{6} +(0.813343 + 1.05997i) q^{7} +(2.13753 - 2.13753i) q^{8} +(-0.381728 - 1.42463i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 512 q - 16 q^{2} - 4 q^{3} - 12 q^{5} - 4 q^{6} - 4 q^{7} + 16 q^{8} - 28 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.62967	−	1.62967i	−1.15235	−	1.15235i	−0.986080	−	0.166269i	\(-0.946828\pi\)
							−0.166269	−	0.986080i	\(-0.553172\pi\)
\(3\)	0.979757	+	0.751794i	0.565663	+	0.434048i	0.851594	−	0.524202i	\(-0.175636\pi\)
							−0.285931	+	0.958250i	\(0.592303\pi\)
\(5\)	1.53210	−	1.99667i	0.685175	−	0.892937i	−0.313214	−	0.949683i	\(-0.601406\pi\)
							0.998389	+	0.0567455i	\(0.0180724\pi\)
\(7\)	0.813343	+	1.05997i	0.307415	+	0.400631i	0.921355	−	0.388722i	\(-0.127083\pi\)
							−0.613940	+	0.789352i	\(0.710417\pi\)
\(11\)	2.92781	+	1.21274i	0.882768	+	0.365655i	0.777570	−	0.628797i	\(-0.216452\pi\)
							0.105198	+	0.994451i	\(0.466452\pi\)
\(13\)	−5.66448	−	3.27039i	−1.57104	−	0.907042i	−0.996041	−	0.0888902i	\(-0.971668\pi\)
							−0.575002	−	0.818152i	\(-0.694999\pi\)
\(17\)	−1.58109	−	3.80791i	−0.383471	−	0.923553i
							\(19\)	−0.603169	+	2.25106i	−0.138376	+	0.516428i	0.861585	+	0.507614i	\(0.169472\pi\)
−0.999961	+	0.00881408i	\(0.997194\pi\)
\(23\)	−0.746928	−	5.67348i	−0.155745	−	1.18300i	−0.873542	−	0.486749i	\(-0.838183\pi\)
							0.717797	−	0.696253i	\(-0.245151\pi\)
\(29\)	8.54416	+	1.12486i	1.58661	+	0.208881i	0.871595	−	0.490227i	\(-0.163086\pi\)
							0.715016	+	0.699108i	\(0.246420\pi\)
\(31\)	0.964922	+	0.740411i	0.173305	+	0.132982i	0.691772	−	0.722116i	\(-0.256830\pi\)
							−0.518467	+	0.855098i	\(0.673497\pi\)
\(37\)	0.970347	+	0.744574i	0.159524	+	0.122407i	0.685439	−	0.728130i	\(-0.259610\pi\)
							−0.525915	+	0.850537i	\(0.676277\pi\)
\(41\)	−0.0269515	+	0.0650666i	−0.00420911	+	0.0101617i	−0.925970	−	0.377597i	\(-0.876751\pi\)
							0.921761	+	0.387758i	\(0.126751\pi\)
\(43\)	2.85493	−	5.90334i	0.435372	−	0.900251i
							\(47\)		−	0.375454i		−	0.0547656i	−0.999625	−	0.0273828i	\(-0.991283\pi\)
0.999625	−	0.0273828i	\(-0.00871731\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.v.a.36.7	✓	512
17.9	even	8	inner	731.2.v.a.638.7	yes	512
43.6	even	3	inner	731.2.v.a.393.7	yes	512
731.264	even	24	inner	731.2.v.a.264.7	yes	512

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.v.a.36.7	✓	512	1.1	even	1	trivial
731.2.v.a.264.7	yes	512	731.264	even	24	inner
731.2.v.a.393.7	yes	512	43.6	even	3	inner
731.2.v.a.638.7	yes	512	17.9	even	8	inner