Embedded newform 731.2.y.a.4.12

• Label: 731.2.y.a.4.12
• Level: $731$
• Weight: $2$
• Character: 731.4
• Analytic conductor: $5.837$
• Analytic rank: $0$
• Dimension: $768$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 731 = 17 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	731.y (of order \(28\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.83706438776\)
Analytic rank:	\(0\)
Dimension:	\(768\)
Relative dimension:	\(64\) over \(\Q(\zeta_{28})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{28}]$

Embedding invariants

Embedding label			4.12
Character	\(\chi\)	\(=\)	731.4
Dual form			731.2.y.a.183.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.57336 - 1.25472i) q^{2} +(-1.09566 - 0.123452i) q^{3} +(0.456119 + 1.99839i) q^{4} +(1.15086 - 0.402703i) q^{5} +(1.56898 + 1.56898i) q^{6} +(-0.279078 - 0.279078i) q^{7} +(0.0434659 - 0.0902579i) q^{8} +(-1.73955 - 0.397040i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 768 q - 14 q^{3} + 104 q^{4} - 6 q^{5} - 36 q^{6} - 28 q^{7}+O(q^{10}) \)

Character values

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

\(n\)	\(173\)	\(562\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{2}{7}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.57336	−	1.25472i	−1.11254	−	0.887218i	−0.118146	−	0.992996i	\(-0.537695\pi\)
							−0.994390	+	0.105779i	\(0.966266\pi\)
\(3\)	−1.09566	−	0.123452i	−0.632582	−	0.0712748i	−0.210149	−	0.977669i	\(-0.567395\pi\)
							−0.422432	+	0.906394i	\(0.638824\pi\)
\(5\)	1.15086	−	0.402703i	0.514680	−	0.180094i	−0.0604267	−	0.998173i	\(-0.519246\pi\)
							0.575107	+	0.818078i	\(0.304960\pi\)
\(7\)	−0.279078	−	0.279078i	−0.105482	−	0.105482i	0.652396	−	0.757878i	\(-0.273764\pi\)
							−0.757878	+	0.652396i	\(0.773764\pi\)
\(11\)	0.761603	−	0.478547i	0.229632	−	0.144287i	−0.412302	−	0.911047i	\(-0.635275\pi\)
							0.641934	+	0.766760i	\(0.278132\pi\)
\(13\)	2.03780	+	0.981353i	0.565184	+	0.272178i	0.694581	−	0.719415i	\(-0.255590\pi\)
							−0.129397	+	0.991593i	\(0.541304\pi\)
\(17\)	−3.89856	−	1.34211i	−0.945539	−	0.325509i
							\(19\)	−0.848850	+	0.193744i	−0.194740	+	0.0444480i	−0.318778	−	0.947829i	\(-0.603272\pi\)
0.124039	+	0.992277i	\(0.460415\pi\)
\(23\)	−4.03960	+	2.53825i	−0.842316	+	0.529262i	−0.882708	−	0.469922i	\(-0.844282\pi\)
							0.0403922	+	0.999184i	\(0.487139\pi\)
\(29\)	0.851582	+	7.55800i	0.158135	+	1.40349i	0.782645	+	0.622469i	\(0.213870\pi\)
							−0.624510	+	0.781017i	\(0.714701\pi\)
\(31\)	−0.398979	−	3.54104i	−0.0716587	−	0.635989i	−0.977288	−	0.211918i	\(-0.932029\pi\)
							0.905629	−	0.424071i	\(-0.139399\pi\)
\(37\)	−4.39637	+	4.39637i	−0.722759	+	0.722759i	−0.969166	−	0.246407i	\(-0.920750\pi\)
							0.246407	+	0.969166i	\(0.420750\pi\)
\(41\)	−4.39473	+	0.495168i	−0.686342	+	0.0773322i	−0.448247	−	0.893910i	\(-0.647952\pi\)
							−0.238095	+	0.971242i	\(0.576523\pi\)
\(43\)	5.77338	+	3.10935i	0.880433	+	0.474171i
							\(47\)	−0.378070	−	1.65643i	−0.0551471	−	0.241615i	0.939840	−	0.341614i	\(-0.110973\pi\)
−0.994988	+	0.0999982i	\(0.968116\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.y.a.4.12	✓	768
17.13	even	4	inner	731.2.y.a.47.53	yes	768
43.11	even	7	inner	731.2.y.a.140.53	yes	768
731.183	even	28	inner	731.2.y.a.183.12	yes	768

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.y.a.4.12	✓	768	1.1	even	1	trivial
731.2.y.a.47.53	yes	768	17.13	even	4	inner
731.2.y.a.140.53	yes	768	43.11	even	7	inner
731.2.y.a.183.12	yes	768	731.183	even	28	inner