Embedded newform 29.8.e.a.4.13

• Label: 29.8.e.a.4.13
• 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			4.13
Character	\(\chi\)	\(=\)	29.4
Dual form			29.8.e.a.22.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(10.9235 + 2.49321i) q^{2} +(-26.6467 - 55.3324i) q^{3} +(-2.21759 - 1.06794i) q^{4} +(-50.3658 + 220.667i) q^{5} +(-153.119 - 670.858i) q^{6} +(-425.854 + 205.081i) q^{7} +(-1142.84 - 911.381i) q^{8} +(-988.053 + 1238.98i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q - 7 q^{2} - 7 q^{3} + 709 q^{4} + 191 q^{5} + 1801 q^{6} - 667 q^{7} + 9086 q^{8} + 9103 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	10.9235	+	2.49321i	0.965509	+	0.220371i	0.676080	−	0.736828i	\(-0.263677\pi\)
							0.289429	+	0.957199i	\(0.406535\pi\)
\(3\)	−26.6467	−	55.3324i	−0.569795	−	1.18319i	−0.964426	−	0.264351i	\(-0.914842\pi\)
							0.394632	−	0.918839i	\(-0.370872\pi\)
\(5\)	−50.3658	+	220.667i	−0.180194	+	0.789482i	0.801342	+	0.598206i	\(0.204120\pi\)
							−0.981536	+	0.191276i	\(0.938737\pi\)
\(7\)	−425.854	+	205.081i	−0.469265	+	0.225986i	−0.653545	−	0.756888i	\(-0.726719\pi\)
							0.184280	+	0.982874i	\(0.441005\pi\)
\(11\)	−5208.82	+	4153.90i	−1.17995	+	0.940982i	−0.999095	−	0.0425274i	\(-0.986459\pi\)
							−0.180858	+	0.983509i	\(0.557888\pi\)
\(13\)	−2537.12	−	3181.44i	−0.320287	−	0.401627i	0.595459	−	0.803386i	\(-0.296970\pi\)
							−0.915745	+	0.401759i	\(0.868399\pi\)
\(17\)		−	11770.6i		−	0.581069i	−0.956865	−	0.290534i	\(-0.906167\pi\)
							0.956865	−	0.290534i	\(-0.0938331\pi\)
\(19\)	12962.8	−	26917.5i	0.433571	−	0.900320i	−0.563663	−	0.826005i	\(-0.690608\pi\)
							0.997235	−	0.0743158i	\(-0.0236773\pi\)
\(23\)	−1205.10	−	5279.89i	−0.0206527	−	0.0904852i	0.963551	−	0.267524i	\(-0.0862054\pi\)
							−0.984204	+	0.177039i	\(0.943348\pi\)
\(29\)	−69718.8	−	111307.i	−0.530831	−	0.847477i
							\(31\)	83859.4	+	19140.4i	0.505575	+	0.115394i	0.467698	−	0.883888i	\(-0.345083\pi\)
0.0378769	+	0.999282i	\(0.487941\pi\)
\(37\)	139965.	+	111618.i	0.454268	+	0.362267i	0.823733	−	0.566978i	\(-0.191888\pi\)
							−0.369465	+	0.929245i	\(0.620459\pi\)
\(41\)		−	370398.i		−	0.839316i	−0.907682	−	0.419658i	\(-0.862150\pi\)
							0.907682	−	0.419658i	\(-0.137850\pi\)
\(43\)	−463217.	+	105726.i	−0.888475	+	0.202789i	−0.642315	−	0.766441i	\(-0.722026\pi\)
							−0.246160	+	0.969229i	\(0.579169\pi\)
\(47\)	−1.01238e6	+	807346.i	−1.42233	+	1.13427i	−0.452137	+	0.891949i	\(0.649338\pi\)
							−0.970196	+	0.242323i	\(0.922091\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.8.e.a.4.13	✓	96
29.22	even	14	inner	29.8.e.a.22.13	yes	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.8.e.a.4.13	✓	96	1.1	even	1	trivial
29.8.e.a.22.13	yes	96	29.22	even	14	inner