Embedded newform 731.2.be.a.59.14

• Label: 731.2.be.a.59.14
• Level: $731$
• Weight: $2$
• Character: 731.59
• Analytic conductor: $5.837$
• Analytic rank: $0$
• Dimension: $1536$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 731 = 17 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	731.be (of order \(56\), degree \(24\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.83706438776\)
Analytic rank:	\(0\)
Dimension:	\(1536\)
Relative dimension:	\(64\) over \(\Q(\zeta_{56})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{56}]$

Embedding invariants

Embedding label			59.14
Character	\(\chi\)	\(=\)	731.59
Dual form			731.2.be.a.508.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.49819 - 0.941376i) q^{2} +(1.42757 + 1.01291i) q^{3} +(0.490620 + 1.01878i) q^{4} +(1.15114 + 3.99569i) q^{5} +(-1.18524 - 2.86142i) q^{6} +(-3.49084 + 1.44595i) q^{7} +(-0.172204 + 1.52835i) q^{8} +(0.0211196 + 0.0603563i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1536 q - 20 q^{2} - 20 q^{3} - 28 q^{5} - 48 q^{6} - 48 q^{7} - 36 q^{8} - 44 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(173\)	\(562\)
\(\chi(n)\)	\(e\left(\frac{5}{8}\right)\)	\(e\left(\frac{4}{7}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.49819	−	0.941376i	−1.05938	−	0.665653i	−0.114619	−	0.993410i	\(-0.536565\pi\)
							−0.944762	+	0.327756i	\(0.893707\pi\)
\(3\)	1.42757	+	1.01291i	0.824207	+	0.584806i	0.911464	−	0.411380i	\(-0.134953\pi\)
							−0.0872571	+	0.996186i	\(0.527810\pi\)
\(5\)	1.15114	+	3.99569i	0.514804	+	1.78693i	0.611078	+	0.791570i	\(0.290736\pi\)
							−0.0962739	+	0.995355i	\(0.530692\pi\)
\(7\)	−3.49084	+	1.44595i	−1.31941	+	0.546518i	−0.927616	−	0.373534i	\(-0.878146\pi\)
							−0.391795	+	0.920052i	\(0.628146\pi\)
\(11\)	−0.107743	+	1.91855i	−0.0324858	+	0.578463i	0.939373	+	0.342898i	\(0.111408\pi\)
							−0.971859	+	0.235565i	\(0.924306\pi\)
\(13\)	0.421462	−	0.336105i	0.116893	−	0.0932188i	−0.563299	−	0.826253i	\(-0.690468\pi\)
							0.680192	+	0.733034i	\(0.261897\pi\)
\(17\)	−4.12130	−	0.122036i	−0.999562	−	0.0295982i
							\(19\)	2.05083	−	5.86093i	0.470492	−	1.34459i	−0.427276	−	0.904121i	\(-0.640527\pi\)
0.897768	−	0.440468i	\(-0.145188\pi\)
\(23\)	−0.326658	+	5.81668i	−0.0681128	+	1.21286i	0.759100	+	0.650974i	\(0.225639\pi\)
							−0.827213	+	0.561888i	\(0.810075\pi\)
\(29\)	−0.732199	+	4.30942i	−0.135966	+	0.800239i	0.833042	+	0.553210i	\(0.186597\pi\)
							−0.969008	+	0.247029i	\(0.920546\pi\)
\(31\)	2.90112	+	0.492920i	0.521056	+	0.0885310i	0.422080	−	0.906559i	\(-0.361300\pi\)
							0.0989765	+	0.995090i	\(0.468443\pi\)
\(37\)	1.12789	−	2.72297i	0.185424	−	0.447654i	−0.803644	−	0.595110i	\(-0.797109\pi\)
							0.989069	+	0.147456i	\(0.0471085\pi\)
\(41\)	−6.54608	−	9.22584i	−1.02233	−	1.44083i	−0.893444	−	0.449174i	\(-0.851718\pi\)
							−0.128882	−	0.991660i	\(-0.541139\pi\)
\(43\)	−6.51649	+	0.731700i	−0.993755	+	0.111583i
							\(47\)	−2.59698	−	5.39269i	−0.378809	−	0.786605i	−0.999996	−	0.00294221i	\(-0.999063\pi\)
0.621187	−	0.783663i	\(-0.286651\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.be.a.59.14	✓	1536
17.15	even	8	inner	731.2.be.a.661.51	yes	1536
43.35	even	7	inner	731.2.be.a.637.51	yes	1536
731.508	even	56	inner	731.2.be.a.508.14	yes	1536

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.be.a.59.14	✓	1536	1.1	even	1	trivial
731.2.be.a.508.14	yes	1536	731.508	even	56	inner
731.2.be.a.637.51	yes	1536	43.35	even	7	inner
731.2.be.a.661.51	yes	1536	17.15	even	8	inner