Embedded newform 29.8.e.a.4.16

• Label: 29.8.e.a.4.16
• Level: $29$
• Weight: $8$
• Character: 29.4
• Analytic conductor: $9.059$
• Analytic rank: $0$
• Dimension: $96$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 29 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	29.e (of order \(14\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.05916573904\)
Analytic rank:	\(0\)
Dimension:	\(96\)
Relative dimension:	\(16\) over \(\Q(\zeta_{14})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{14}]$

Embedding invariants

Embedding label			4.16
Character	\(\chi\)	\(=\)	29.4
Dual form			29.8.e.a.22.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(19.1958 + 4.38132i) q^{2} +(-23.2918 - 48.3658i) q^{3} +(233.959 + 112.669i) q^{4} +(36.5819 - 160.276i) q^{5} +(-235.198 - 1030.47i) q^{6} +(541.526 - 260.785i) q^{7} +(2026.99 + 1616.47i) q^{8} +(-433.175 + 543.185i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q - 7 q^{2} - 7 q^{3} + 709 q^{4} + 191 q^{5} + 1801 q^{6} - 667 q^{7} + 9086 q^{8} + 9103 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{14}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	19.1958	+	4.38132i	1.69669	+	0.387257i	0.957995	−	0.286786i	\(-0.0925870\pi\)
							0.738691	+	0.674044i	\(0.235444\pi\)
\(3\)	−23.2918	−	48.3658i	−0.498056	−	1.03422i	−0.986822	−	0.161809i	\(-0.948267\pi\)
							0.488767	−	0.872415i	\(-0.337447\pi\)
\(5\)	36.5819	−	160.276i	0.130879	−	0.573420i	−0.866377	−	0.499391i	\(-0.833557\pi\)
							0.997256	−	0.0740289i	\(-0.0235857\pi\)
\(7\)	541.526	−	260.785i	0.596728	−	0.287369i	−0.111038	−	0.993816i	\(-0.535418\pi\)
							0.707766	+	0.706447i	\(0.249703\pi\)
\(11\)	2129.34	−	1698.09i	0.482360	−	0.384669i	−0.351903	−	0.936037i	\(-0.614465\pi\)
							0.834262	+	0.551368i	\(0.185894\pi\)
\(13\)	−1921.80	−	2409.86i	−0.242608	−	0.304221i	0.645588	−	0.763686i	\(-0.276613\pi\)
							−0.888196	+	0.459465i	\(0.848041\pi\)
\(17\)			37148.8i			1.83389i	0.399014	+	0.916945i	\(0.369352\pi\)
							−0.399014	+	0.916945i	\(0.630648\pi\)
\(19\)	−18984.2	+	39421.2i	−0.634974	+	1.31854i	0.296606	+	0.955000i	\(0.404145\pi\)
							−0.931580	+	0.363537i	\(0.881569\pi\)
\(23\)	−7058.45	−	30925.1i	−0.120966	−	0.529985i	−0.998706	−	0.0508490i	\(-0.983807\pi\)
							0.877741	−	0.479136i	\(-0.159050\pi\)
\(29\)	−10685.2	+	130903.i	−0.0813561	+	0.996685i
							\(31\)	3778.37	+	862.388i	0.0227792	+	0.00519920i	0.233895	−	0.972262i	\(-0.424853\pi\)
−0.211116	+	0.977461i	\(0.567710\pi\)
\(37\)	158260.	+	126208.i	0.513647	+	0.409620i	0.845715	−	0.533636i	\(-0.179175\pi\)
							−0.332067	+	0.943256i	\(0.607746\pi\)
\(41\)		−	718435.i		−	1.62796i	−0.580892	−	0.813981i	\(-0.697296\pi\)
							0.580892	−	0.813981i	\(-0.302704\pi\)
\(43\)	278766.	−	63626.5i	0.534688	−	0.122039i	0.0533528	−	0.998576i	\(-0.483009\pi\)
							0.481335	+	0.876537i	\(0.340152\pi\)
\(47\)	−495480.	+	395132.i	−0.696119	+	0.555136i	−0.906356	−	0.422514i	\(-0.861148\pi\)
							0.210238	+	0.977650i	\(0.432576\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.8.e.a.4.16	✓	96
29.22	even	14	inner	29.8.e.a.22.16	yes	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.8.e.a.4.16	✓	96	1.1	even	1	trivial
29.8.e.a.22.16	yes	96	29.22	even	14	inner