Embedded newform 29.3.f.a.21.1

• Label: 29.3.f.a.21.1
• Level: $29$
• Weight: $3$
• Character: 29.21
• 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			21.1
Character	\(\chi\)	\(=\)	29.21
Dual form			29.3.f.a.18.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.23928 + 3.54166i) q^{2} +(4.35885 + 0.491124i) q^{3} +(-7.88021 - 6.28426i) q^{4} +(-0.825141 - 1.71342i) q^{5} +(-7.14123 + 14.8289i) q^{6} +(-2.97342 - 3.72855i) q^{7} +(19.3141 - 12.1359i) q^{8} +(9.98399 + 2.27878i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 16 q^{2} - 12 q^{3} - 14 q^{4} - 14 q^{5} - 14 q^{6} - 10 q^{7} + 28 q^{8} - 14 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{17}{28}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.23928	+	3.54166i	−0.619640	+	1.77083i	0.0172211	+	0.999852i	\(0.494518\pi\)
							−0.636861	+	0.770978i	\(0.719768\pi\)
\(3\)	4.35885	+	0.491124i	1.45295	+	0.163708i	0.802868	−	0.596157i	\(-0.203306\pi\)
							0.650081	+	0.759865i	\(0.274735\pi\)
\(5\)	−0.825141	−	1.71342i	−0.165028	−	0.342685i	0.802012	−	0.597308i	\(-0.203763\pi\)
							−0.967040	+	0.254623i	\(0.918049\pi\)
\(7\)	−2.97342	−	3.72855i	−0.424774	−	0.532649i	0.522686	−	0.852525i	\(-0.324930\pi\)
							−0.947459	+	0.319876i	\(0.896359\pi\)
\(11\)	8.99985	+	5.65498i	0.818169	+	0.514089i	0.874902	−	0.484300i	\(-0.160926\pi\)
							−0.0567334	+	0.998389i	\(0.518069\pi\)
\(13\)	−14.5195	+	3.31399i	−1.11689	+	0.254922i	−0.740842	−	0.671679i	\(-0.765573\pi\)
							−0.376045	+	0.926601i	\(0.622716\pi\)
\(17\)	−1.89819	+	1.89819i	−0.111658	+	0.111658i	−0.760728	−	0.649070i	\(-0.775158\pi\)
							0.649070	+	0.760728i	\(0.275158\pi\)
\(19\)	−0.860526	−	7.63738i	−0.0452908	−	0.401967i	−0.995820	−	0.0913331i	\(-0.970887\pi\)
							0.950530	−	0.310634i	\(-0.100541\pi\)
\(23\)	−12.8995	−	6.21208i	−0.560849	−	0.270090i	0.131908	−	0.991262i	\(-0.457890\pi\)
							−0.692757	+	0.721172i	\(0.743604\pi\)
\(29\)	−13.6496	+	25.5869i	−0.470674	+	0.882307i
							\(31\)	−5.75026	+	16.4333i	−0.185492	+	0.530107i	−0.998650	−	0.0519363i	\(-0.983461\pi\)
0.813158	+	0.582043i	\(0.197746\pi\)
\(37\)	47.2178	−	29.6689i	1.27616	−	0.801862i	0.288346	−	0.957526i	\(-0.406895\pi\)
							0.987810	+	0.155664i	\(0.0497518\pi\)
\(41\)	52.6192	+	52.6192i	1.28339	+	1.28339i	0.938724	+	0.344671i	\(0.112010\pi\)
							0.344671	+	0.938724i	\(0.387990\pi\)
\(43\)	−57.3351	+	20.0624i	−1.33338	+	0.466568i	−0.900488	−	0.434880i	\(-0.856791\pi\)
							−0.432887	+	0.901448i	\(0.642505\pi\)
\(47\)	−14.1131	+	22.4608i	−0.300278	+	0.477889i	−0.962449	−	0.271463i	\(-0.912492\pi\)
							0.662171	+	0.749353i	\(0.269635\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.3.f.a.21.1	yes	48
3.2	odd	2		261.3.s.a.253.4		48
29.18	odd	28	inner	29.3.f.a.18.1	✓	48
87.47	even	28		261.3.s.a.163.4		48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.3.f.a.18.1	✓	48	29.18	odd	28	inner
29.3.f.a.21.1	yes	48	1.1	even	1	trivial
261.3.s.a.163.4		48	87.47	even	28
261.3.s.a.253.4		48	3.2	odd	2