Embedded newform 731.2.y.a.4.20

• Label: 731.2.y.a.4.20
• 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			4.20
Character	\(\chi\)	\(=\)	731.4
Dual form			731.2.y.a.183.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.921645 - 0.734988i) q^{2} +(-0.106119 - 0.0119568i) q^{3} +(-0.135818 - 0.595060i) q^{4} +(0.980725 - 0.343171i) q^{5} +(0.0890163 + 0.0890163i) q^{6} +(1.87363 + 1.87363i) q^{7} +(-1.33513 + 2.77243i) q^{8} +(-2.91367 - 0.665025i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 768 q - 14 q^{3} + 104 q^{4} - 6 q^{5} - 36 q^{6} - 28 q^{7}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.921645	−	0.734988i	−0.651702	−	0.519715i	0.240906	−	0.970548i	\(-0.422555\pi\)
							−0.892608	+	0.450834i	\(0.851127\pi\)
\(3\)	−0.106119	−	0.0119568i	−0.0612680	−	0.00690325i	0.0812773	−	0.996692i	\(-0.474100\pi\)
							−0.142545	+	0.989788i	\(0.545529\pi\)
\(5\)	0.980725	−	0.343171i	0.438594	−	0.153471i	−0.101944	−	0.994790i	\(-0.532506\pi\)
							0.540538	+	0.841320i	\(0.318221\pi\)
\(7\)	1.87363	+	1.87363i	0.708165	+	0.708165i	0.966149	−	0.257984i	\(-0.0830582\pi\)
							−0.257984	+	0.966149i	\(0.583058\pi\)
\(11\)	3.00990	−	1.89125i	0.907520	−	0.570232i	0.00457514	−	0.999990i	\(-0.498544\pi\)
							0.902945	+	0.429757i	\(0.141401\pi\)
\(13\)	−3.71868	−	1.79082i	−1.03138	−	0.496685i	−0.159906	−	0.987132i	\(-0.551119\pi\)
							−0.871471	+	0.490448i	\(0.836833\pi\)
\(17\)	−1.69835	−	3.75708i	−0.411910	−	0.911225i
							\(19\)	3.28426	−	0.749611i	0.753461	−	0.171972i	0.171494	−	0.985185i	\(-0.445141\pi\)
0.581967	+	0.813213i	\(0.302283\pi\)
\(23\)	6.15890	−	3.86989i	1.28422	−	0.806928i	0.295303	−	0.955404i	\(-0.404579\pi\)
							0.988916	+	0.148475i	\(0.0474366\pi\)
\(29\)	−0.377922	−	3.35415i	−0.0701784	−	0.622850i	−0.978741	−	0.205101i	\(-0.934248\pi\)
							0.908562	−	0.417749i	\(-0.137181\pi\)
\(31\)	−0.790413	−	7.01511i	−0.141962	−	1.25995i	−0.839483	−	0.543386i	\(-0.817142\pi\)
							0.697521	−	0.716565i	\(-0.254286\pi\)
\(37\)	6.39605	−	6.39605i	1.05150	−	1.05150i	0.0529045	−	0.998600i	\(-0.483152\pi\)
							0.998600	−	0.0529045i	\(-0.0168479\pi\)
\(41\)	−11.3103	+	1.27437i	−1.76637	+	0.199023i	−0.934505	−	0.355950i	\(-0.884157\pi\)
							−0.831869	+	0.554972i	\(0.812729\pi\)
\(43\)	−5.67274	+	3.28938i	−0.865085	+	0.501625i
							\(47\)	1.14624	+	5.02202i	0.167197	+	0.732537i	0.987109	+	0.160048i	\(0.0511649\pi\)
−0.819913	+	0.572489i	\(0.805978\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.y.a.4.20	✓	768
17.13	even	4	inner	731.2.y.a.47.45	yes	768
43.11	even	7	inner	731.2.y.a.140.45	yes	768
731.183	even	28	inner	731.2.y.a.183.20	yes	768

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.y.a.4.20	✓	768	1.1	even	1	trivial
731.2.y.a.47.45	yes	768	17.13	even	4	inner
731.2.y.a.140.45	yes	768	43.11	even	7	inner
731.2.y.a.183.20	yes	768	731.183	even	28	inner