Embedded newform 29.3.f.a.19.1

• Label: 29.3.f.a.19.1
• Level: $29$
• Weight: $3$
• Character: 29.19
• 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			19.1
Character	\(\chi\)	\(=\)	29.19
Dual form			29.3.f.a.26.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.63282 - 2.59861i) q^{2} +(0.532023 - 1.52043i) q^{3} +(-2.35117 + 4.88225i) q^{4} +(-4.43970 - 1.01333i) q^{5} +(-4.81972 + 1.10007i) q^{6} +(10.7635 - 5.18344i) q^{7} +(4.32723 - 0.487562i) q^{8} +(5.00781 + 3.99360i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 16 q^{2} - 12 q^{3} - 14 q^{4} - 14 q^{5} - 14 q^{6} - 10 q^{7} + 28 q^{8} - 14 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{9}{28}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.63282	−	2.59861i	−0.816409	−	1.29931i	−0.951244	−	0.308439i	\(-0.900193\pi\)
							0.134835	−	0.990868i	\(-0.456950\pi\)
\(3\)	0.532023	−	1.52043i	0.177341	−	0.506811i	−0.820561	−	0.571559i	\(-0.806339\pi\)
							0.997902	+	0.0647482i	\(0.0206244\pi\)
\(5\)	−4.43970	−	1.01333i	−0.887940	−	0.202667i	−0.245861	−	0.969305i	\(-0.579071\pi\)
							−0.642079	+	0.766638i	\(0.721928\pi\)
\(7\)	10.7635	−	5.18344i	1.53765	−	0.740491i	0.542607	−	0.839986i	\(-0.317437\pi\)
							0.995038	+	0.0994954i	\(0.0317229\pi\)
\(11\)	2.90150	+	0.326920i	0.263772	+	0.0297200i	0.242860	−	0.970061i	\(-0.421914\pi\)
							0.0209122	+	0.999781i	\(0.493343\pi\)
\(13\)	2.24911	−	1.79360i	0.173008	−	0.137969i	−0.533158	−	0.846016i	\(-0.678995\pi\)
							0.706166	+	0.708046i	\(0.250423\pi\)
\(17\)	2.44971	−	2.44971i	0.144101	−	0.144101i	−0.631376	−	0.775477i	\(-0.717510\pi\)
							0.775477	+	0.631376i	\(0.217510\pi\)
\(19\)	−31.9711	+	11.1872i	−1.68269	+	0.588799i	−0.991056	−	0.133450i	\(-0.957395\pi\)
							−0.691634	+	0.722248i	\(0.743109\pi\)
\(23\)	9.24886	+	40.5219i	0.402124	+	1.76182i	0.618771	+	0.785571i	\(0.287631\pi\)
							−0.216647	+	0.976250i	\(0.569512\pi\)
\(29\)	18.6826	−	22.1802i	0.644226	−	0.764835i
							\(31\)	7.40308	+	11.7819i	0.238809	+	0.380063i	0.944529	−	0.328429i	\(-0.106519\pi\)
−0.705720	+	0.708491i	\(0.749376\pi\)
\(37\)	−19.6990	+	2.21954i	−0.532405	+	0.0599876i	−0.374075	−	0.927399i	\(-0.622040\pi\)
							−0.158330	+	0.987386i	\(0.550611\pi\)
\(41\)	−15.2314	−	15.2314i	−0.371497	−	0.371497i	0.496526	−	0.868022i	\(-0.334609\pi\)
							−0.868022	+	0.496526i	\(0.834609\pi\)
\(43\)	1.58568	+	0.996350i	0.0368763	+	0.0231709i	0.550344	−	0.834938i	\(-0.314497\pi\)
							−0.513468	+	0.858109i	\(0.671639\pi\)
\(47\)	−0.858173	+	7.61650i	−0.0182590	+	0.162053i	−0.999533	−	0.0305651i	\(-0.990269\pi\)
							0.981274	+	0.192618i	\(0.0616979\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.3.f.a.19.1	✓	48
3.2	odd	2		261.3.s.a.19.4		48
29.26	odd	28	inner	29.3.f.a.26.1	yes	48
87.26	even	28		261.3.s.a.55.4		48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.3.f.a.19.1	✓	48	1.1	even	1	trivial
29.3.f.a.26.1	yes	48	29.26	odd	28	inner
261.3.s.a.19.4		48	3.2	odd	2
261.3.s.a.55.4		48	87.26	even	28