Embedded newform 29.3.f.a.26.1

• Label: 29.3.f.a.26.1
• Level: $29$
• Weight: $3$
• Character: 29.26
• 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			26.1
Character	\(\chi\)	\(=\)	29.26
Dual form			29.3.f.a.19.1

$q$-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 \(\left(\mathbb{Z}/29\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{19}{28}\right)\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.3.f.a.19.1	✓	48	29.19	odd	28	inner
29.3.f.a.26.1	yes	48	1.1	even	1	trivial
261.3.s.a.19.4		48	87.77	even	28
261.3.s.a.55.4		48	3.2	odd	2