Embedded newform 731.2.v.a.36.2

• Label: 731.2.v.a.36.2
• 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			36.2
Character	\(\chi\)	\(=\)	731.36
Dual form			731.2.v.a.264.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.88749 - 1.88749i) q^{2} +(-2.07239 - 1.59020i) q^{3} +5.12526i q^{4} +(0.776315 - 1.01171i) q^{5} +(0.910131 + 6.91313i) q^{6} +(-2.46818 - 3.21659i) q^{7} +(5.89891 - 5.89891i) q^{8} +(0.989611 + 3.69328i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 512 q - 16 q^{2} - 4 q^{3} - 12 q^{5} - 4 q^{6} - 4 q^{7} + 16 q^{8} - 28 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.88749	−	1.88749i	−1.33466	−	1.33466i	−0.901148	−	0.433511i	\(-0.857275\pi\)
							−0.433511	−	0.901148i	\(-0.642725\pi\)
\(3\)	−2.07239	−	1.59020i	−1.19650	−	0.918105i	−0.198422	−	0.980117i	\(-0.563582\pi\)
							−0.998075	+	0.0620121i	\(0.980248\pi\)
\(5\)	0.776315	−	1.01171i	0.347179	−	0.452452i	−0.586826	−	0.809713i	\(-0.699623\pi\)
							0.934005	+	0.357261i	\(0.116289\pi\)
\(7\)	−2.46818	−	3.21659i	−0.932883	−	1.21576i	−0.976239	−	0.216697i	\(-0.930471\pi\)
							0.0433556	−	0.999060i	\(-0.486195\pi\)
\(11\)	0.174624	+	0.0723318i	0.0526512	+	0.0218089i	0.408854	−	0.912600i	\(-0.365929\pi\)
							−0.356202	+	0.934409i	\(0.615929\pi\)
\(13\)	−4.92483	−	2.84335i	−1.36590	−	0.788604i	−0.375500	−	0.926822i	\(-0.622529\pi\)
							−0.990402	+	0.138219i	\(0.955862\pi\)
\(17\)	−3.34433	+	2.41152i	−0.811120	+	0.584879i
							\(19\)	0.543431	−	2.02811i	0.124672	−	0.465281i	−0.875156	−	0.483840i	\(-0.839241\pi\)
0.999828	+	0.0185600i	\(0.00590816\pi\)
\(23\)	0.0349635	+	0.265574i	0.00729040	+	0.0553761i	0.994703	−	0.102787i	\(-0.0327761\pi\)
							−0.987413	+	0.158163i	\(0.949443\pi\)
\(29\)	−5.90196	−	0.777008i	−1.09597	−	0.144287i	−0.439207	−	0.898386i	\(-0.644740\pi\)
							−0.656760	+	0.754099i	\(0.728074\pi\)
\(31\)	1.59263	+	1.22207i	0.286045	+	0.219490i	0.741810	−	0.670610i	\(-0.233968\pi\)
							−0.455765	+	0.890100i	\(0.650634\pi\)
\(37\)	3.63482	+	2.78910i	0.597561	+	0.458525i	0.862721	−	0.505680i	\(-0.168758\pi\)
							−0.265160	+	0.964204i	\(0.585425\pi\)
\(41\)	2.17745	−	5.25683i	0.340061	−	0.820979i	−0.657648	−	0.753325i	\(-0.728449\pi\)
							0.997709	−	0.0676539i	\(-0.0215514\pi\)
\(43\)	1.49452	−	6.38486i	0.227912	−	0.973682i
							\(47\)		−	4.66772i		−	0.680857i	−0.940270	−	0.340428i	\(-0.889428\pi\)
0.940270	−	0.340428i	\(-0.110572\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.v.a.36.2	✓	512
17.9	even	8	inner	731.2.v.a.638.2	yes	512
43.6	even	3	inner	731.2.v.a.393.2	yes	512
731.264	even	24	inner	731.2.v.a.264.2	yes	512

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.v.a.36.2	✓	512	1.1	even	1	trivial
731.2.v.a.264.2	yes	512	731.264	even	24	inner
731.2.v.a.393.2	yes	512	43.6	even	3	inner
731.2.v.a.638.2	yes	512	17.9	even	8	inner