Embedded newform 29.3.f.a.14.2

• Label: 29.3.f.a.14.2
• Level: $29$
• Weight: $3$
• Character: 29.14
• 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			14.2
Character	\(\chi\)	\(=\)	29.14
Dual form			29.3.f.a.27.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.170607 - 1.51418i) q^{2} +(2.08647 + 3.32060i) q^{3} +(1.63608 - 0.373424i) q^{4} +(-5.91405 - 4.71630i) q^{5} +(4.67202 - 3.72581i) q^{6} +(-1.42870 + 6.25955i) q^{7} +(-2.85762 - 8.16662i) q^{8} +(-2.76806 + 5.74794i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 16 q^{2} - 12 q^{3} - 14 q^{4} - 14 q^{5} - 14 q^{6} - 10 q^{7} + 28 q^{8} - 14 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{13}{28}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.170607	−	1.51418i	−0.0853036	−	0.757090i	−0.961438	−	0.275023i	\(-0.911315\pi\)
							0.876134	−	0.482067i	\(-0.160114\pi\)
\(3\)	2.08647	+	3.32060i	0.695490	+	1.10687i	0.988568	+	0.150778i	\(0.0481777\pi\)
							−0.293077	+	0.956089i	\(0.594679\pi\)
\(5\)	−5.91405	−	4.71630i	−1.18281	−	0.943260i	−0.183600	−	0.983001i	\(-0.558775\pi\)
							−0.999210	+	0.0397411i	\(0.987347\pi\)
\(7\)	−1.42870	+	6.25955i	−0.204100	+	0.894222i	0.764308	+	0.644852i	\(0.223081\pi\)
							−0.968408	+	0.249370i	\(0.919776\pi\)
\(11\)	−5.33282	+	15.2403i	−0.484802	+	1.38548i	0.398710	+	0.917077i	\(0.369458\pi\)
							−0.883512	+	0.468408i	\(0.844828\pi\)
\(13\)	−5.47235	−	11.3635i	−0.420950	−	0.874112i	−0.998337	−	0.0576417i	\(-0.981642\pi\)
							0.577387	−	0.816471i	\(-0.304072\pi\)
\(17\)	11.5542	−	11.5542i	0.679659	−	0.679659i	−0.280264	−	0.959923i	\(-0.590422\pi\)
							0.959923	+	0.280264i	\(0.0904219\pi\)
\(19\)	12.1000	+	7.60291i	0.636840	+	0.400153i	0.811398	−	0.584494i	\(-0.198707\pi\)
							−0.174558	+	0.984647i	\(0.555850\pi\)
\(23\)	−2.84328	−	3.56536i	−0.123621	−	0.155016i	0.716170	−	0.697926i	\(-0.245894\pi\)
							−0.839791	+	0.542910i	\(0.817322\pi\)
\(29\)	23.5053	+	16.9853i	0.810527	+	0.585702i
							\(31\)	−1.32747	−	11.7816i	−0.0428217	−	0.380053i	−0.996785	−	0.0801282i	\(-0.974467\pi\)
0.953963	−	0.299925i	\(-0.0969616\pi\)
\(37\)	2.68961	+	7.68647i	0.0726922	+	0.207742i	0.974497	−	0.224402i	\(-0.0720428\pi\)
							−0.901804	+	0.432144i	\(0.857757\pi\)
\(41\)	10.8682	+	10.8682i	0.265077	+	0.265077i	0.827113	−	0.562036i	\(-0.189982\pi\)
							−0.562036	+	0.827113i	\(0.689982\pi\)
\(43\)	−41.8092	−	4.71076i	−0.972307	−	0.109553i	−0.388503	−	0.921447i	\(-0.627008\pi\)
							−0.583804	+	0.811895i	\(0.698436\pi\)
\(47\)	−28.2097	−	9.87101i	−0.600207	−	0.210022i	0.0130369	−	0.999915i	\(-0.495850\pi\)
							−0.613244	+	0.789893i	\(0.710136\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.3.f.a.14.2	✓	48
3.2	odd	2		261.3.s.a.217.3		48
29.27	odd	28	inner	29.3.f.a.27.2	yes	48
87.56	even	28		261.3.s.a.172.3		48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.3.f.a.14.2	✓	48	1.1	even	1	trivial
29.3.f.a.27.2	yes	48	29.27	odd	28	inner
261.3.s.a.172.3		48	87.56	even	28
261.3.s.a.217.3		48	3.2	odd	2