Embedded newform 731.2.v.a.36.13

• Label: 731.2.v.a.36.13
• 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			36.13
Character	\(\chi\)	\(=\)	731.36
Dual form			731.2.v.a.264.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.31498 - 1.31498i) q^{2} +(2.01865 + 1.54896i) q^{3} +1.45837i q^{4} +(-0.221616 + 0.288815i) q^{5} +(-0.617629 - 4.69135i) q^{6} +(1.79020 + 2.33304i) q^{7} +(-0.712236 + 0.712236i) q^{8} +(0.899197 + 3.35585i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 512 q - 16 q^{2} - 4 q^{3} - 12 q^{5} - 4 q^{6} - 4 q^{7} + 16 q^{8} - 28 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.31498	−	1.31498i	−0.929835	−	0.929835i	0.0678603	−	0.997695i	\(-0.478383\pi\)
							−0.997695	+	0.0678603i	\(0.978383\pi\)
\(3\)	2.01865	+	1.54896i	1.16547	+	0.894294i	0.995831	−	0.0912124i	\(-0.0290742\pi\)
							0.169636	+	0.985507i	\(0.445741\pi\)
\(5\)	−0.221616	+	0.288815i	−0.0991096	+	0.129162i	−0.840275	−	0.542160i	\(-0.817607\pi\)
							0.741166	+	0.671322i	\(0.234273\pi\)
\(7\)	1.79020	+	2.33304i	0.676633	+	0.881805i	0.997873	−	0.0651928i	\(-0.0207662\pi\)
							−0.321240	+	0.946998i	\(0.604100\pi\)
\(11\)	−4.01558	−	1.66331i	−1.21074	−	0.501506i	−0.316287	−	0.948663i	\(-0.602436\pi\)
							−0.894455	+	0.447157i	\(0.852436\pi\)
\(13\)	3.60611	+	2.08199i	1.00016	+	0.577440i	0.908294	−	0.418332i	\(-0.137385\pi\)
							0.0918612	+	0.995772i	\(0.470718\pi\)
\(17\)	−0.653877	−	4.07093i	−0.158588	−	0.987345i
							\(19\)	−1.27563	+	4.76071i	−0.292649	+	1.09218i	0.650417	+	0.759578i	\(0.274594\pi\)
−0.943066	+	0.332605i	\(0.892072\pi\)
\(23\)	0.525824	+	3.99403i	0.109642	+	0.832813i	0.954581	+	0.297951i	\(0.0963031\pi\)
							−0.844939	+	0.534862i	\(0.820364\pi\)
\(29\)	4.35561	+	0.573427i	0.808817	+	0.106483i	0.523583	−	0.851975i	\(-0.324595\pi\)
							0.285235	+	0.958458i	\(0.407929\pi\)
\(31\)	2.17207	+	1.66669i	0.390116	+	0.299346i	0.785133	−	0.619328i	\(-0.212595\pi\)
							−0.395017	+	0.918674i	\(0.629261\pi\)
\(37\)	−7.51465	−	5.76619i	−1.23540	−	0.947956i	−0.235685	−	0.971829i	\(-0.575733\pi\)
							−0.999715	+	0.0238735i	\(0.992400\pi\)
\(41\)	−1.37316	+	3.31509i	−0.214451	+	0.517730i	−0.994098	−	0.108489i	\(-0.965399\pi\)
							0.779647	+	0.626220i	\(0.215399\pi\)
\(43\)	−2.54725	+	6.04248i	−0.388452	+	0.921469i
							\(47\)			5.55428i			0.810174i	0.914278	+	0.405087i	\(0.132759\pi\)
−0.914278	+	0.405087i	\(0.867241\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.v.a.36.13	✓	512
17.9	even	8	inner	731.2.v.a.638.13	yes	512
43.6	even	3	inner	731.2.v.a.393.13	yes	512
731.264	even	24	inner	731.2.v.a.264.13	yes	512

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.v.a.36.13	✓	512	1.1	even	1	trivial
731.2.v.a.264.13	yes	512	731.264	even	24	inner
731.2.v.a.393.13	yes	512	43.6	even	3	inner
731.2.v.a.638.13	yes	512	17.9	even	8	inner