Embedded newform 731.2.n.a.208.7

• Label: 731.2.n.a.208.7
• Level: $731$
• Weight: $2$
• Character: 731.208
• 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			208.7
Character	\(\chi\)	\(=\)	731.208
Dual form			731.2.n.a.608.58

$q$-expansion

\(f(q)\)	\(=\)	\(q-2.28700i q^{2} +(1.18102 + 0.316454i) q^{3} -3.23035 q^{4} +(-0.126627 - 0.0339295i) q^{5} +(0.723730 - 2.70100i) q^{6} +(-4.66179 + 1.24912i) q^{7} +2.81381i q^{8} +(-1.30340 - 0.752519i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q - 6 q^{3} - 264 q^{4} + 2 q^{5} - 2 q^{6}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)		−	2.28700i		−	1.61715i	−0.588393	−	0.808575i	\(-0.700239\pi\)
							0.588393	−	0.808575i	\(-0.299761\pi\)
\(3\)	1.18102	+	0.316454i	0.681865	+	0.182705i	0.583093	−	0.812405i	\(-0.301842\pi\)
							0.0987712	+	0.995110i	\(0.468509\pi\)
\(5\)	−0.126627	−	0.0339295i	−0.0566291	−	0.0151737i	0.230393	−	0.973098i	\(-0.425999\pi\)
							−0.287022	+	0.957924i	\(0.592665\pi\)
\(7\)	−4.66179	+	1.24912i	−1.76199	+	0.472124i	−0.987118	−	0.159992i	\(-0.948853\pi\)
							−0.774873	+	0.632116i	\(0.782186\pi\)
\(11\)	4.02891	+	4.02891i	1.21476	+	1.21476i	0.969442	+	0.245321i	\(0.0788935\pi\)
							0.245321	+	0.969442i	\(0.421107\pi\)
\(13\)	−3.19478	+	5.53352i	−0.886073	+	1.53472i	−0.0415932	+	0.999135i	\(0.513243\pi\)
							−0.844479	+	0.535588i	\(0.820090\pi\)
\(17\)	1.31468	+	3.90789i	0.318857	+	0.947803i
							\(19\)	−2.72681	+	1.57433i	−0.625574	+	0.361175i	−0.779036	−	0.626979i	\(-0.784291\pi\)
0.153462	+	0.988155i	\(0.450958\pi\)
\(23\)	1.24482	−	4.64571i	0.259562	−	0.968698i	−0.705934	−	0.708278i	\(-0.749472\pi\)
							0.965495	−	0.260420i	\(-0.0838610\pi\)
\(29\)	−2.01987	−	7.53825i	−0.375080	−	1.39982i	−0.853228	−	0.521539i	\(-0.825358\pi\)
							0.478147	−	0.878280i	\(-0.341308\pi\)
\(31\)	2.23828	+	0.599745i	0.402007	+	0.107717i	0.454157	−	0.890922i	\(-0.349941\pi\)
							−0.0521498	+	0.998639i	\(0.516607\pi\)
\(37\)	−8.74117	−	2.34219i	−1.43704	−	0.385053i	−0.545543	−	0.838083i	\(-0.683677\pi\)
							−0.891496	+	0.453029i	\(0.850343\pi\)
\(41\)	5.31793	+	5.31793i	0.830521	+	0.830521i	0.987588	−	0.157067i	\(-0.0502037\pi\)
							−0.157067	+	0.987588i	\(0.550204\pi\)
\(43\)	−2.53194	+	6.04891i	−0.386118	+	0.922450i
							\(47\)	1.76469			0.257407			0.128703	−	0.991683i	\(-0.458918\pi\)
0.128703	+	0.991683i	\(0.458918\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.n.a.208.7	✓	256
17.13	even	4	inner	731.2.n.a.251.58	yes	256
43.6	even	3	inner	731.2.n.a.565.7	yes	256
731.608	even	12	inner	731.2.n.a.608.58	yes	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.n.a.208.7	✓	256	1.1	even	1	trivial
731.2.n.a.251.58	yes	256	17.13	even	4	inner
731.2.n.a.565.7	yes	256	43.6	even	3	inner
731.2.n.a.608.58	yes	256	731.608	even	12	inner