Embedded newform 29.3.f.a.8.2

• Label: 29.3.f.a.8.2
• 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			8.2
Character	\(\chi\)	\(=\)	29.8
Dual form			29.3.f.a.11.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.20133 - 0.770280i) q^{2} +(-0.552425 + 4.90291i) q^{3} +(1.12521 + 0.897324i) q^{4} +(2.64264 + 5.48750i) q^{5} +(4.99268 - 10.3674i) q^{6} +(-3.22936 - 4.04949i) q^{7} +(3.17747 + 5.05691i) q^{8} +(-14.9590 - 3.41428i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 16 q^{2} - 12 q^{3} - 14 q^{4} - 14 q^{5} - 14 q^{6} - 10 q^{7} + 28 q^{8} - 14 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.20133	−	0.770280i	−1.10067	−	0.385140i	−0.282003	−	0.959414i	\(-0.590999\pi\)
							−0.818663	+	0.574274i	\(0.805285\pi\)
\(3\)	−0.552425	+	4.90291i	−0.184142	+	1.63430i	0.473375	+	0.880861i	\(0.343036\pi\)
							−0.657516	+	0.753440i	\(0.728393\pi\)
\(5\)	2.64264	+	5.48750i	0.528528	+	1.09750i	0.978840	+	0.204628i	\(0.0655985\pi\)
							−0.450312	+	0.892871i	\(0.648687\pi\)
\(7\)	−3.22936	−	4.04949i	−0.461337	−	0.578499i	0.495689	−	0.868500i	\(-0.334916\pi\)
							−0.957026	+	0.290001i	\(0.906344\pi\)
\(11\)	3.16517	−	5.03734i	0.287743	−	0.457940i	−0.671306	−	0.741180i	\(-0.734266\pi\)
							0.959049	+	0.283240i	\(0.0914093\pi\)
\(13\)	14.6839	−	3.35151i	1.12953	−	0.257808i	0.383385	−	0.923589i	\(-0.374758\pi\)
							0.746148	+	0.665780i	\(0.231901\pi\)
\(17\)	22.0898	+	22.0898i	1.29940	+	1.29940i	0.928787	+	0.370615i	\(0.120853\pi\)
							0.370615	+	0.928787i	\(0.379147\pi\)
\(19\)	−0.835449	+	0.0941325i	−0.0439710	+	0.00495434i	−0.133923	−	0.990992i	\(-0.542757\pi\)
							0.0899516	+	0.995946i	\(0.471329\pi\)
\(23\)	−21.1449	−	10.1828i	−0.919344	−	0.442733i	−0.0865068	−	0.996251i	\(-0.527570\pi\)
							−0.832837	+	0.553519i	\(0.813285\pi\)
\(29\)	−18.7362	−	22.1350i	−0.646074	−	0.763274i
							\(31\)	21.3525	+	7.47156i	0.688790	+	0.241018i	0.651904	−	0.758301i	\(-0.273970\pi\)
0.0368860	+	0.999319i	\(0.488256\pi\)
\(37\)	4.25720	+	6.77529i	0.115059	+	0.183116i	0.899289	−	0.437355i	\(-0.144085\pi\)
							−0.784230	+	0.620470i	\(0.786942\pi\)
\(41\)	−8.24028	+	8.24028i	−0.200982	+	0.200982i	−0.800421	−	0.599438i	\(-0.795391\pi\)
							0.599438	+	0.800421i	\(0.295391\pi\)
\(43\)	−3.09691	−	8.85047i	−0.0720212	−	0.205825i	0.902246	−	0.431222i	\(-0.141918\pi\)
							−0.974267	+	0.225398i	\(0.927632\pi\)
\(47\)	−22.1022	−	13.8877i	−0.470259	−	0.295483i	0.275978	−	0.961164i	\(-0.410998\pi\)
							−0.746237	+	0.665681i	\(0.768141\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.3.f.a.8.2	✓	48
3.2	odd	2		261.3.s.a.37.3		48
29.11	odd	28	inner	29.3.f.a.11.2	yes	48
87.11	even	28		261.3.s.a.127.3		48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.3.f.a.8.2	✓	48	1.1	even	1	trivial
29.3.f.a.11.2	yes	48	29.11	odd	28	inner
261.3.s.a.37.3		48	3.2	odd	2
261.3.s.a.127.3		48	87.11	even	28