Embedded newform 29.6.e.a.13.8

• Label: 29.6.e.a.13.8
• Level: $29$
• Weight: $6$
• Character: 29.13
• Analytic conductor: $4.651$
• Analytic rank: $0$
• Dimension: $72$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(4.65113077458\)
Analytic rank:	\(0\)
Dimension:	\(72\)
Relative dimension:	\(12\) over \(\Q(\zeta_{14})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{14}]$

Embedding invariants

Embedding label			13.8
Character	\(\chi\)	\(=\)	29.13
Dual form			29.6.e.a.9.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.714414 - 1.48350i) q^{2} +(-7.48323 - 5.96767i) q^{3} +(18.2613 + 22.8989i) q^{4} +(37.6666 + 18.1393i) q^{5} +(-14.1991 + 6.83795i) q^{6} +(122.778 - 153.959i) q^{7} +(98.3854 - 22.4558i) q^{8} +(-33.6870 - 147.593i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 7 q^{2} - 7 q^{3} + 165 q^{4} - 53 q^{5} - 311 q^{6} - 27 q^{7} - 1498 q^{8} + 1567 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{9}{14}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.714414	−	1.48350i	0.126292	−	0.262248i	−0.828230	−	0.560388i	\(-0.810652\pi\)
							0.954522	+	0.298140i	\(0.0963663\pi\)
\(3\)	−7.48323	−	5.96767i	−0.480049	−	0.382826i	0.353356	−	0.935489i	\(-0.385040\pi\)
							−0.833405	+	0.552663i	\(0.813612\pi\)
\(5\)	37.6666	+	18.1393i	0.673801	+	0.324485i	0.739309	−	0.673367i	\(-0.235153\pi\)
							−0.0655080	+	0.997852i	\(0.520867\pi\)
\(7\)	122.778	−	153.959i	0.947059	−	1.18757i	−0.0350743	−	0.999385i	\(-0.511167\pi\)
							0.982133	−	0.188189i	\(-0.0602618\pi\)
\(11\)	353.360	+	80.6522i	0.880514	+	0.200971i	0.638797	−	0.769375i	\(-0.279432\pi\)
							0.241717	+	0.970347i	\(0.422290\pi\)
\(13\)	−205.015	+	898.227i	−0.336454	+	1.47410i	0.469926	+	0.882706i	\(0.344281\pi\)
							−0.806380	+	0.591397i	\(0.798576\pi\)
\(17\)		−	500.479i		−	0.420014i	−0.977700	−	0.210007i	\(-0.932651\pi\)
							0.977700	−	0.210007i	\(-0.0673487\pi\)
\(19\)	1207.46	−	962.918i	0.767342	−	0.611935i	−0.159582	−	0.987185i	\(-0.551015\pi\)
							0.926924	+	0.375250i	\(0.122443\pi\)
\(23\)	−3821.87	+	1840.51i	−1.50645	+	0.725470i	−0.991299	−	0.131627i	\(-0.957980\pi\)
							−0.515155	+	0.857097i	\(0.672266\pi\)
\(29\)	−3646.65	−	2685.72i	−0.805191	−	0.593015i
							\(31\)	626.496	−	1300.93i	0.117088	−	0.243137i	−0.834185	−	0.551484i	\(-0.814062\pi\)
0.951274	+	0.308348i	\(0.0997760\pi\)
\(37\)	−1955.26	+	446.275i	−0.234801	+	0.0535918i	−0.338302	−	0.941038i	\(-0.609853\pi\)
							0.103501	+	0.994629i	\(0.466995\pi\)
\(41\)		−	4862.03i		−	0.451709i	−0.974161	−	0.225854i	\(-0.927483\pi\)
							0.974161	−	0.225854i	\(-0.0725173\pi\)
\(43\)	9555.25	+	19841.7i	0.788081	+	1.63647i	0.771186	+	0.636609i	\(0.219664\pi\)
							0.0168943	+	0.999857i	\(0.494622\pi\)
\(47\)	−26108.1	−	5959.01i	−1.72397	−	0.393486i	−0.758027	−	0.652224i	\(-0.773836\pi\)
							−0.965948	+	0.258738i	\(0.916693\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.6.e.a.13.8	yes	72
29.3	odd	28		841.6.a.l.1.42		72
29.9	even	14	inner	29.6.e.a.9.8	✓	72
29.26	odd	28		841.6.a.l.1.31		72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.6.e.a.9.8	✓	72	29.9	even	14	inner
29.6.e.a.13.8	yes	72	1.1	even	1	trivial
841.6.a.l.1.31		72	29.26	odd	28
841.6.a.l.1.42		72	29.3	odd	28