Embedded newform 731.2.y.a.4.4

• Label: 731.2.y.a.4.4
• 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			4.4
Character	\(\chi\)	\(=\)	731.4
Dual form			731.2.y.a.183.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.00227 - 1.59676i) q^{2} +(3.12141 + 0.351699i) q^{3} +(1.01442 + 4.44445i) q^{4} +(-2.71628 + 0.950466i) q^{5} +(-5.68835 - 5.68835i) q^{6} +(-0.896379 - 0.896379i) q^{7} +(2.84322 - 5.90401i) q^{8} +(6.69474 + 1.52803i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 768 q - 14 q^{3} + 104 q^{4} - 6 q^{5} - 36 q^{6} - 28 q^{7}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.00227	−	1.59676i	−1.41582	−	1.12908i	−0.972550	−	0.232693i	\(-0.925246\pi\)
							−0.443271	−	0.896387i	\(-0.646182\pi\)
\(3\)	3.12141	+	0.351699i	1.80215	+	0.203053i	0.948280	−	0.317435i	\(-0.102822\pi\)
							0.853869	+	0.520489i	\(0.174250\pi\)
\(5\)	−2.71628	+	0.950466i	−1.21476	+	0.425061i	−0.860192	−	0.509970i	\(-0.829656\pi\)
							−0.354564	+	0.935032i	\(0.615371\pi\)
\(7\)	−0.896379	−	0.896379i	−0.338799	−	0.338799i	0.517116	−	0.855915i	\(-0.327006\pi\)
							−0.855915	+	0.517116i	\(0.827006\pi\)
\(11\)	−3.53846	+	2.22336i	−1.06689	+	0.670369i	−0.946613	−	0.322372i	\(-0.895520\pi\)
							−0.120272	+	0.992741i	\(0.538377\pi\)
\(13\)	0.684148	+	0.329468i	0.189749	+	0.0913781i	0.526346	−	0.850271i	\(-0.323562\pi\)
							−0.336597	+	0.941649i	\(0.609276\pi\)
\(17\)	0.393017	+	4.10433i	0.0953207	+	0.995447i
							\(19\)	−4.50603	+	1.02847i	−1.03375	+	0.235947i	−0.705550	−	0.708660i	\(-0.749300\pi\)
−0.328203	+	0.944607i	\(0.606443\pi\)
\(23\)	−2.73781	+	1.72028i	−0.570872	+	0.358703i	−0.786326	−	0.617811i	\(-0.788020\pi\)
							0.215454	+	0.976514i	\(0.430877\pi\)
\(29\)	−0.361945	−	3.21235i	−0.0672115	−	0.596518i	−0.981501	−	0.191457i	\(-0.938679\pi\)
							0.914290	−	0.405061i	\(-0.132750\pi\)
\(31\)	1.06660	+	9.46632i	0.191567	+	1.70020i	0.610363	+	0.792122i	\(0.291024\pi\)
							−0.418796	+	0.908080i	\(0.637548\pi\)
\(37\)	5.89312	−	5.89312i	0.968824	−	0.968824i	−0.0307048	−	0.999528i	\(-0.509775\pi\)
							0.999528	+	0.0307048i	\(0.00977518\pi\)
\(41\)	−11.8411	+	1.33417i	−1.84927	+	0.208363i	−0.965651	−	0.259842i	\(-0.916329\pi\)
							−0.883620	+	0.468205i	\(0.844901\pi\)
\(43\)	−2.61210	+	6.01473i	−0.398342	+	0.917237i
							\(47\)	−0.538776	−	2.36053i	−0.0785885	−	0.344319i	0.920313	−	0.391183i	\(-0.127934\pi\)
−0.998901	+	0.0468646i	\(0.985077\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.y.a.4.4	✓	768
17.13	even	4	inner	731.2.y.a.47.61	yes	768
43.11	even	7	inner	731.2.y.a.140.61	yes	768
731.183	even	28	inner	731.2.y.a.183.4	yes	768

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.y.a.4.4	✓	768	1.1	even	1	trivial
731.2.y.a.47.61	yes	768	17.13	even	4	inner
731.2.y.a.140.61	yes	768	43.11	even	7	inner
731.2.y.a.183.4	yes	768	731.183	even	28	inner