Embedded newform 731.2.n.a.608.60

• Label: 731.2.n.a.608.60
• Level: $731$
• Weight: $2$
• Character: 731.608
• Analytic conductor: $5.837$
• Analytic rank: $0$
• Dimension: $256$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 731 = 17 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	731.n (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.83706438776\)
Analytic rank:	\(0\)
Dimension:	\(256\)
Relative dimension:	\(64\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			608.60
Character	\(\chi\)	\(=\)	731.608
Dual form			731.2.n.a.208.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+2.53263i q^{2} +(-2.42847 + 0.650707i) q^{3} -4.41419 q^{4} +(-2.11001 + 0.565376i) q^{5} +(-1.64800 - 6.15041i) q^{6} +(-4.08395 - 1.09429i) q^{7} -6.11424i q^{8} +(2.87598 - 1.66045i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q - 6 q^{3} - 264 q^{4} + 2 q^{5} - 2 q^{6}+O(q^{10}) \)

Character values

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

\(n\)	\(173\)	\(562\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)			2.53263i			1.79084i	0.445226	+	0.895418i	\(0.353123\pi\)
							−0.445226	+	0.895418i	\(0.646877\pi\)
\(3\)	−2.42847	+	0.650707i	−1.40208	+	0.375686i	−0.879091	−	0.476654i	\(-0.841850\pi\)
							−0.522988	+	0.852340i	\(0.675183\pi\)
\(5\)	−2.11001	+	0.565376i	−0.943627	+	0.252844i	−0.697656	−	0.716433i	\(-0.745773\pi\)
							−0.245971	+	0.969277i	\(0.579107\pi\)
\(7\)	−4.08395	−	1.09429i	−1.54359	−	0.413603i	−0.616165	−	0.787617i	\(-0.711315\pi\)
							−0.927423	+	0.374014i	\(0.877981\pi\)
\(11\)	−1.07177	+	1.07177i	−0.323152	+	0.323152i	−0.849975	−	0.526823i	\(-0.823383\pi\)
							0.526823	+	0.849975i	\(0.323383\pi\)
\(13\)	1.97616	+	3.42281i	0.548088	+	0.949317i	0.998406	+	0.0564483i	\(0.0179776\pi\)
							−0.450317	+	0.892869i	\(0.648689\pi\)
\(17\)	−3.54610	+	2.10361i	−0.860055	+	0.510202i
							\(19\)	−3.99000	−	2.30363i	−0.915369	−	0.528488i	−0.0332140	−	0.999448i	\(-0.510574\pi\)
−0.882155	+	0.470960i	\(0.843908\pi\)
\(23\)	0.742606	+	2.77144i	0.154844	+	0.577886i	0.999119	+	0.0419750i	\(0.0133650\pi\)
							−0.844275	+	0.535911i	\(0.819968\pi\)
\(29\)	−2.45006	+	9.14374i	−0.454964	+	1.69795i	0.233228	+	0.972422i	\(0.425071\pi\)
							−0.688192	+	0.725528i	\(0.741596\pi\)
\(31\)	−8.63854	+	2.31469i	−1.55153	+	0.415731i	−0.929969	−	0.367637i	\(-0.880167\pi\)
							−0.621559	+	0.783368i	\(0.713500\pi\)
\(37\)	7.30442	−	1.95721i	1.20084	−	0.321764i	0.397678	−	0.917525i	\(-0.369816\pi\)
							0.803162	+	0.595761i	\(0.203149\pi\)
\(41\)	−1.37910	+	1.37910i	−0.215379	+	0.215379i	−0.806548	−	0.591169i	\(-0.798667\pi\)
							0.591169	+	0.806548i	\(0.298667\pi\)
\(43\)	1.01850	−	6.47786i	0.155320	−	0.987864i
							\(47\)	6.20009			0.904376			0.452188	−	0.891923i	\(-0.350644\pi\)
0.452188	+	0.891923i	\(0.350644\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.n.a.608.60	yes	256
17.4	even	4	inner	731.2.n.a.565.5	yes	256
43.36	even	3	inner	731.2.n.a.251.60	yes	256
731.208	even	12	inner	731.2.n.a.208.5	✓	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.n.a.208.5	✓	256	731.208	even	12	inner
731.2.n.a.251.60	yes	256	43.36	even	3	inner
731.2.n.a.565.5	yes	256	17.4	even	4	inner
731.2.n.a.608.60	yes	256	1.1	even	1	trivial