Embedded newform 731.2.s.b.386.16

• Label: 731.2.s.b.386.16
• Level: $731$
• Weight: $2$
• Character: 731.386
• Analytic conductor: $5.837$
• Analytic rank: $0$
• Dimension: $496$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 731 = 17 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	731.s (of order \(16\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.83706438776\)
Analytic rank:	\(0\)
Dimension:	\(496\)
Relative dimension:	\(62\) over \(\Q(\zeta_{16})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{16}]$

Embedding invariants

Embedding label			386.16
Character	\(\chi\)	\(=\)	731.386
Dual form			731.2.s.b.214.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.35638 + 0.561829i) q^{2} +(-0.901163 - 1.34869i) q^{3} +(0.109890 - 0.109890i) q^{4} +(-0.614380 + 3.08870i) q^{5} +(1.98005 + 1.32302i) q^{6} +(0.564221 + 2.83653i) q^{7} +(1.03635 - 2.50196i) q^{8} +(0.141192 - 0.340868i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 496 q - 16 q^{4} - 16 q^{6} - 16 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(173\)	\(562\)
\(\chi(n)\)	\(e\left(\frac{13}{16}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.35638	+	0.561829i	−0.959103	+	0.397273i	−0.806645	−	0.591037i	\(-0.798719\pi\)
							−0.152458	+	0.988310i	\(0.548719\pi\)
\(3\)	−0.901163	−	1.34869i	−0.520287	−	0.778664i	0.474542	−	0.880233i	\(-0.342614\pi\)
							−0.994829	+	0.101569i	\(0.967614\pi\)
\(5\)	−0.614380	+	3.08870i	−0.274759	+	1.38131i	0.558995	+	0.829171i	\(0.311187\pi\)
							−0.833754	+	0.552136i	\(0.813813\pi\)
\(7\)	0.564221	+	2.83653i	0.213256	+	1.07211i	0.927959	+	0.372681i	\(0.121561\pi\)
							−0.714704	+	0.699427i	\(0.753439\pi\)
\(11\)	3.23584	−	4.84277i	0.975641	−	1.46015i	0.0899111	−	0.995950i	\(-0.471342\pi\)
							0.885730	−	0.464200i	\(-0.153658\pi\)
\(13\)	−1.98670	−	1.98670i	−0.551011	−	0.551011i	0.375721	−	0.926733i	\(-0.377395\pi\)
							−0.926733	+	0.375721i	\(0.877395\pi\)
\(17\)	−0.798749	+	4.04500i	−0.193725	+	0.981056i
							\(19\)	3.64965	−	1.51173i	0.837287	−	0.346816i	0.0775040	−	0.996992i	\(-0.475305\pi\)
0.759783	+	0.650176i	\(0.225305\pi\)
\(23\)	2.41523	+	1.61381i	0.503611	+	0.336502i	0.781286	−	0.624173i	\(-0.214564\pi\)
							−0.277676	+	0.960675i	\(0.589564\pi\)
\(29\)	−0.315181	−	0.0626933i	−0.0585276	−	0.0116419i	0.165740	−	0.986170i	\(-0.446999\pi\)
							−0.224267	+	0.974528i	\(0.571999\pi\)
\(31\)	0.229031	+	0.342770i	0.0411353	+	0.0615633i	0.851467	−	0.524408i	\(-0.175713\pi\)
							−0.810332	+	0.585971i	\(0.800713\pi\)
\(37\)	−2.21301	−	3.31201i	−0.363817	−	0.544491i	0.603728	−	0.797191i	\(-0.293681\pi\)
							−0.967545	+	0.252700i	\(0.918681\pi\)
\(41\)	8.17437	−	1.62598i	1.27662	−	0.253936i	0.490191	−	0.871615i	\(-0.336927\pi\)
							0.786430	+	0.617679i	\(0.211927\pi\)
\(43\)	4.31984	+	4.93346i	0.658769	+	0.752345i
							\(47\)	9.15733	+	9.15733i	1.33573	+	1.33573i	0.900147	+	0.435587i	\(0.143459\pi\)
0.435587	+	0.900147i	\(0.356541\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.s.b.386.16	yes	496
17.10	odd	16	inner	731.2.s.b.214.47	yes	496
43.42	odd	2	inner	731.2.s.b.386.47	yes	496
731.214	even	16	inner	731.2.s.b.214.16	✓	496

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.s.b.214.16	✓	496	731.214	even	16	inner
731.2.s.b.214.47	yes	496	17.10	odd	16	inner
731.2.s.b.386.16	yes	496	1.1	even	1	trivial
731.2.s.b.386.47	yes	496	43.42	odd	2	inner