Embedded newform 29.3.f.a.8.3

• Label: 29.3.f.a.8.3
• Level: $29$
• Weight: $3$
• Character: 29.8
• Analytic conductor: $0.790$
• Analytic rank: $0$
• Dimension: $48$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 29 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	29.f (of order \(28\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.790192766645\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Relative dimension:	\(4\) over \(\Q(\zeta_{28})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{28}]$

Embedding invariants

Embedding label			8.3
Character	\(\chi\)	\(=\)	29.8
Dual form			29.3.f.a.11.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0310749 - 0.0108736i) q^{2} +(0.529545 - 4.69985i) q^{3} +(-3.12648 - 2.49328i) q^{4} +(3.79007 + 7.87017i) q^{5} +(-0.0675598 + 0.140289i) q^{6} +(3.62612 + 4.54701i) q^{7} +(0.140107 + 0.222980i) q^{8} +(-13.0338 - 2.97487i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 16 q^{2} - 12 q^{3} - 14 q^{4} - 14 q^{5} - 14 q^{6} - 10 q^{7} + 28 q^{8} - 14 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/29\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{3}{28}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0310749	−	0.0108736i	−0.0155375	−	0.00543680i	0.322499	−	0.946570i	\(-0.395477\pi\)
							−0.338037	+	0.941133i	\(0.609763\pi\)
\(3\)	0.529545	−	4.69985i	0.176515	−	1.56662i	−0.523211	−	0.852203i	\(-0.675266\pi\)
							0.699726	−	0.714412i	\(-0.253306\pi\)
\(5\)	3.79007	+	7.87017i	0.758015	+	1.57403i	0.817584	+	0.575809i	\(0.195313\pi\)
							−0.0595693	+	0.998224i	\(0.518973\pi\)
\(7\)	3.62612	+	4.54701i	0.518017	+	0.649572i	0.970187	−	0.242360i	\(-0.0779214\pi\)
							−0.452170	+	0.891932i	\(0.649350\pi\)
\(11\)	3.95365	−	6.29219i	0.359422	−	0.572017i	−0.617246	−	0.786770i	\(-0.711752\pi\)
							0.976668	+	0.214753i	\(0.0688946\pi\)
\(13\)	−10.7515	+	2.45395i	−0.827037	+	0.188766i	−0.615031	−	0.788503i	\(-0.710857\pi\)
							−0.212006	+	0.977268i	\(0.567999\pi\)
\(17\)	−4.26171	−	4.26171i	−0.250689	−	0.250689i	0.570564	−	0.821253i	\(-0.306725\pi\)
							−0.821253	+	0.570564i	\(0.806725\pi\)
\(19\)	−13.0613	+	1.47165i	−0.687436	+	0.0774554i	−0.448771	−	0.893647i	\(-0.648138\pi\)
							−0.238664	+	0.971102i	\(0.576710\pi\)
\(23\)	−6.32686	−	3.04685i	−0.275081	−	0.132472i	0.291258	−	0.956645i	\(-0.405926\pi\)
							−0.566339	+	0.824173i	\(0.691641\pi\)
\(29\)	29.0000	+	0.0339620i	0.999999	+	0.00117110i
							\(31\)	2.37249	+	0.830170i	0.0765319	+	0.0267797i	0.368273	−	0.929718i	\(-0.379949\pi\)
−0.291742	+	0.956497i	\(0.594235\pi\)
\(37\)	−17.7197	−	28.2007i	−0.478910	−	0.762182i	0.516661	−	0.856190i	\(-0.327175\pi\)
							−0.995571	+	0.0940085i	\(0.970032\pi\)
\(41\)	−2.70371	+	2.70371i	−0.0659440	+	0.0659440i	−0.739310	−	0.673366i	\(-0.764848\pi\)
							0.673366	+	0.739310i	\(0.264848\pi\)
\(43\)	−11.9973	−	34.2863i	−0.279007	−	0.797356i	−0.995057	−	0.0993068i	\(-0.968337\pi\)
							0.716050	−	0.698049i	\(-0.245948\pi\)
\(47\)	7.75262	+	4.87130i	0.164949	+	0.103645i	0.611977	−	0.790876i	\(-0.290375\pi\)
							−0.447027	+	0.894520i	\(0.647517\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.3.f.a.8.3	✓	48
3.2	odd	2		261.3.s.a.37.2		48
29.11	odd	28	inner	29.3.f.a.11.3	yes	48
87.11	even	28		261.3.s.a.127.2		48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.3.f.a.8.3	✓	48	1.1	even	1	trivial
29.3.f.a.11.3	yes	48	29.11	odd	28	inner
261.3.s.a.37.2		48	3.2	odd	2
261.3.s.a.127.2		48	87.11	even	28