Embedded newform 29.3.f.a.2.1

• Label: 29.3.f.a.2.1
• Level: $29$
• Weight: $3$
• Character: 29.2
• 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			2.1
Character	\(\chi\)	\(=\)	29.2
Dual form			29.3.f.a.15.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.56136 - 0.401269i) q^{2} +(1.78374 + 1.12080i) q^{3} +(8.62256 + 1.96804i) q^{4} +(5.24106 - 4.17960i) q^{5} +(-5.90279 - 4.70732i) q^{6} +(2.11977 + 9.28734i) q^{7} +(-16.3872 - 5.73413i) q^{8} +(-1.97942 - 4.11031i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 16 q^{2} - 12 q^{3} - 14 q^{4} - 14 q^{5} - 14 q^{6} - 10 q^{7} + 28 q^{8} - 14 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{28}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.56136	−	0.401269i	−1.78068	−	0.200634i	−0.840616	−	0.541632i	\(-0.817807\pi\)
							−0.940064	+	0.340997i	\(0.889235\pi\)
\(3\)	1.78374	+	1.12080i	0.594579	+	0.373599i	0.795450	−	0.606019i	\(-0.207234\pi\)
							−0.200871	+	0.979618i	\(0.564377\pi\)
\(5\)	5.24106	−	4.17960i	1.04821	−	0.835920i	0.0614515	−	0.998110i	\(-0.480427\pi\)
							0.986760	+	0.162190i	\(0.0518556\pi\)
\(7\)	2.11977	+	9.28734i	0.302825	+	1.32676i	0.865842	+	0.500317i	\(0.166783\pi\)
							−0.563017	+	0.826445i	\(0.690360\pi\)
\(11\)	−0.700768	+	0.245209i	−0.0637061	+	0.0222917i	−0.361944	−	0.932200i	\(-0.617887\pi\)
							0.298238	+	0.954492i	\(0.403601\pi\)
\(13\)	3.39382	−	7.04735i	0.261063	−	0.542104i	−0.728698	−	0.684835i	\(-0.759874\pi\)
							0.989761	+	0.142731i	\(0.0455885\pi\)
\(17\)	−17.4730	+	17.4730i	−1.02782	+	1.02782i	−0.0282209	+	0.999602i	\(0.508984\pi\)
							−0.999602	+	0.0282209i	\(0.991016\pi\)
\(19\)	−7.96846	−	12.6817i	−0.419393	−	0.667460i	0.568485	−	0.822694i	\(-0.307530\pi\)
							−0.987878	+	0.155234i	\(0.950387\pi\)
\(23\)	−11.9732	+	15.0139i	−0.520575	+	0.652780i	−0.970731	−	0.240170i	\(-0.922797\pi\)
							0.450156	+	0.892950i	\(0.351368\pi\)
\(29\)	−19.7681	−	21.2184i	−0.681659	−	0.731670i
							\(31\)	0.327624	+	0.0369143i	0.0105685	+	0.00119078i	0.117247	−	0.993103i	\(-0.462593\pi\)
−0.106679	+	0.994294i	\(0.534022\pi\)
\(37\)	−15.2693	−	5.34296i	−0.412684	−	0.144404i	0.115949	−	0.993255i	\(-0.463009\pi\)
							−0.528632	+	0.848851i	\(0.677295\pi\)
\(41\)	28.0484	+	28.0484i	0.684106	+	0.684106i	0.960923	−	0.276816i	\(-0.0892794\pi\)
							−0.276816	+	0.960923i	\(0.589279\pi\)
\(43\)	0.679170	+	6.02780i	0.0157947	+	0.140181i	0.999135	−	0.0415949i	\(-0.0132439\pi\)
							−0.983340	+	0.181776i	\(0.941815\pi\)
\(47\)	1.43620	+	4.10443i	0.0305575	+	0.0873282i	0.958152	−	0.286261i	\(-0.0924126\pi\)
							−0.927594	+	0.373590i	\(0.878127\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.3.f.a.2.1	✓	48
3.2	odd	2		261.3.s.a.118.4		48
29.15	odd	28	inner	29.3.f.a.15.1	yes	48
87.44	even	28		261.3.s.a.73.4		48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.3.f.a.2.1	✓	48	1.1	even	1	trivial
29.3.f.a.15.1	yes	48	29.15	odd	28	inner
261.3.s.a.73.4		48	87.44	even	28
261.3.s.a.118.4		48	3.2	odd	2