Embedded newform 731.2.s.b.214.1

• Label: 731.2.s.b.214.1
• Level: $731$
• Weight: $2$
• Character: 731.214
• Analytic conductor: $5.837$
• Analytic rank: $0$
• Dimension: $496$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 731 = 17 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	731.s (of order \(16\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.83706438776\)
Analytic rank:	\(0\)
Dimension:	\(496\)
Relative dimension:	\(62\) over \(\Q(\zeta_{16})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{16}]$

Embedding invariants

Embedding label			214.1
Character	\(\chi\)	\(=\)	731.214
Dual form			731.2.s.b.386.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.59137 - 1.07338i) q^{2} +(1.43157 - 2.14250i) q^{3} +(4.14883 + 4.14883i) q^{4} +(0.218697 + 1.09946i) q^{5} +(-6.00945 + 4.01538i) q^{6} +(0.509391 - 2.56088i) q^{7} +(-4.15111 - 10.0217i) q^{8} +(-1.39286 - 3.36265i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 496 q - 16 q^{4} - 16 q^{6} - 16 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(173\)	\(562\)
\(\chi(n)\)	\(e\left(\frac{3}{16}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.59137	−	1.07338i	−1.83237	−	0.758994i	−0.965468	−	0.260520i	\(-0.916106\pi\)
							−0.866905	−	0.498474i	\(-0.833894\pi\)
\(3\)	1.43157	−	2.14250i	0.826519	−	1.23697i	−0.142452	−	0.989802i	\(-0.545499\pi\)
							0.968972	−	0.247172i	\(-0.0795012\pi\)
\(5\)	0.218697	+	1.09946i	0.0978043	+	0.491695i	0.998375	+	0.0569857i	\(0.0181489\pi\)
							−0.900571	+	0.434709i	\(0.856851\pi\)
\(7\)	0.509391	−	2.56088i	0.192532	−	0.967923i	−0.756800	−	0.653647i	\(-0.773238\pi\)
							0.949332	−	0.314276i	\(-0.101762\pi\)
\(11\)	−3.00828	−	4.50220i	−0.907029	−	1.35747i	−0.933789	−	0.357824i	\(-0.883519\pi\)
							0.0267599	−	0.999642i	\(-0.491481\pi\)
\(13\)	2.57289	−	2.57289i	0.713592	−	0.713592i	−0.253693	−	0.967285i	\(-0.581645\pi\)
							0.967285	+	0.253693i	\(0.0816452\pi\)
\(17\)	0.322379	+	4.11048i	0.0781884	+	0.996939i
							\(19\)	0.0971443	+	0.0402385i	0.0222864	+	0.00923134i	0.393799	−	0.919197i	\(-0.371161\pi\)
−0.371512	+	0.928428i	\(0.621161\pi\)
\(23\)	4.04501	−	2.70279i	0.843443	−	0.563570i	−0.0570850	−	0.998369i	\(-0.518181\pi\)
							0.900528	+	0.434799i	\(0.143181\pi\)
\(29\)	−3.76314	+	0.748536i	−0.698798	+	0.139000i	−0.531690	−	0.846939i	\(-0.678443\pi\)
							−0.167108	+	0.985939i	\(0.553443\pi\)
\(31\)	−1.58349	+	2.36987i	−0.284404	+	0.425641i	−0.945974	−	0.324243i	\(-0.894890\pi\)
							0.661570	+	0.749884i	\(0.269890\pi\)
\(37\)	−0.297333	+	0.444991i	−0.0488813	+	0.0731560i	−0.855105	−	0.518454i	\(-0.826508\pi\)
							0.806224	+	0.591610i	\(0.201508\pi\)
\(41\)	−4.26906	−	0.849169i	−0.666716	−	0.132618i	−0.149882	−	0.988704i	\(-0.547889\pi\)
							−0.516834	+	0.856086i	\(0.672889\pi\)
\(43\)	−3.73871	−	5.38721i	−0.570148	−	0.821542i
							\(47\)	−6.53596	+	6.53596i	−0.953368	+	0.953368i	−0.998960	−	0.0455921i	\(-0.985483\pi\)
0.0455921	+	0.998960i	\(0.485483\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.s.b.214.1	✓	496
17.12	odd	16	inner	731.2.s.b.386.62	yes	496
43.42	odd	2	inner	731.2.s.b.214.62	yes	496
731.386	even	16	inner	731.2.s.b.386.1	yes	496

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.s.b.214.1	✓	496	1.1	even	1	trivial
731.2.s.b.214.62	yes	496	43.42	odd	2	inner
731.2.s.b.386.1	yes	496	731.386	even	16	inner
731.2.s.b.386.62	yes	496	17.12	odd	16	inner