Embedded newform 29.6.e.a.13.3

• Label: 29.6.e.a.13.3
• 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			13.3
Character	\(\chi\)	\(=\)	29.13
Dual form			29.6.e.a.9.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.94013 + 6.10524i) q^{2} +(18.0943 + 14.4298i) q^{3} +(-8.67795 - 10.8818i) q^{4} +(71.3244 + 34.3480i) q^{5} +(-141.297 + 68.0450i) q^{6} +(126.722 - 158.905i) q^{7} +(-119.455 + 27.2648i) q^{8} +(65.1147 + 285.286i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 7 q^{2} - 7 q^{3} + 165 q^{4} - 53 q^{5} - 311 q^{6} - 27 q^{7} - 1498 q^{8} + 1567 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.94013	+	6.10524i	−0.519746	+	1.07926i	0.461613	+	0.887082i	\(0.347271\pi\)
							−0.981359	+	0.192183i	\(0.938443\pi\)
\(3\)	18.0943	+	14.4298i	1.16075	+	0.925669i	0.998137	−	0.0610204i	\(-0.0194355\pi\)
							0.162616	+	0.986689i	\(0.448007\pi\)
\(5\)	71.3244	+	34.3480i	1.27589	+	0.614436i	0.944330	−	0.328999i	\(-0.106711\pi\)
							0.331559	+	0.943435i	\(0.392426\pi\)
\(7\)	126.722	−	158.905i	0.977482	−	1.22572i	0.00329128	−	0.999995i	\(-0.498952\pi\)
							0.974190	−	0.225728i	\(-0.0724762\pi\)
\(11\)	−631.223	−	144.073i	−1.57290	−	0.359004i	−0.654941	−	0.755680i	\(-0.727307\pi\)
							−0.917959	+	0.396675i	\(0.870164\pi\)
\(13\)	87.5650	−	383.647i	0.143705	−	0.629613i	−0.850851	−	0.525408i	\(-0.823913\pi\)
							0.994556	−	0.104206i	\(-0.0332300\pi\)
\(17\)		−	1162.81i		−	0.975861i	−0.872882	−	0.487931i	\(-0.837752\pi\)
							0.872882	−	0.487931i	\(-0.162248\pi\)
\(19\)	556.551	−	443.834i	0.353688	−	0.282057i	−0.430486	−	0.902597i	\(-0.641658\pi\)
							0.784175	+	0.620540i	\(0.213087\pi\)
\(23\)	−2650.61	+	1276.47i	−1.04478	+	0.503141i	−0.875899	−	0.482495i	\(-0.839731\pi\)
							−0.168884	+	0.985636i	\(0.554016\pi\)
\(29\)	1470.23	+	4283.64i	0.324632	+	0.945840i
							\(31\)	−986.932	+	2049.39i	−0.184452	+	0.383018i	−0.972606	−	0.232460i	\(-0.925323\pi\)
0.788154	+	0.615478i	\(0.211037\pi\)
\(37\)	−10039.6	+	2291.48i	−1.20563	+	0.275177i	−0.777701	−	0.628635i	\(-0.783614\pi\)
							−0.427930	+	0.903812i	\(0.640757\pi\)
\(41\)			7386.01i			0.686199i	0.939299	+	0.343100i	\(0.111477\pi\)
							−0.939299	+	0.343100i	\(0.888523\pi\)
\(43\)	1338.80	+	2780.06i	0.110420	+	0.229289i	0.948855	−	0.315713i	\(-0.102244\pi\)
							−0.838435	+	0.545001i	\(0.816529\pi\)
\(47\)	8790.44	+	2006.36i	0.580452	+	0.132484i	0.502658	−	0.864485i	\(-0.332355\pi\)
							0.0777934	+	0.996970i	\(0.475213\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.6.e.a.13.3	yes	72
29.3	odd	28		841.6.a.l.1.18		72
29.9	even	14	inner	29.6.e.a.9.3	✓	72
29.26	odd	28		841.6.a.l.1.55		72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.6.e.a.9.3	✓	72	29.9	even	14	inner
29.6.e.a.13.3	yes	72	1.1	even	1	trivial
841.6.a.l.1.18		72	29.3	odd	28
841.6.a.l.1.55		72	29.26	odd	28