Embedded newform 29.6.e.a.22.1

• Label: 29.6.e.a.22.1
• Level: $29$
• Weight: $6$
• Character: 29.22
• 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			22.1
Character	\(\chi\)	\(=\)	29.22
Dual form			29.6.e.a.4.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-10.9467 + 2.49852i) q^{2} +(-2.67824 + 5.56142i) q^{3} +(84.7573 - 40.8170i) q^{4} +(-8.95546 - 39.2364i) q^{5} +(15.4226 - 67.5710i) q^{6} +(52.7478 + 25.4020i) q^{7} +(-544.918 + 434.558i) q^{8} +(127.752 + 160.195i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 7 q^{2} - 7 q^{3} + 165 q^{4} - 53 q^{5} - 311 q^{6} - 27 q^{7} - 1498 q^{8} + 1567 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{13}{14}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−10.9467	+	2.49852i	−1.93513	+	0.441680i	−0.940264	+	0.340446i	\(0.889422\pi\)
							−0.994863	+	0.101234i	\(0.967721\pi\)
\(3\)	−2.67824	+	5.56142i	−0.171809	+	0.356765i	−0.969038	−	0.246912i	\(-0.920584\pi\)
							0.797229	+	0.603677i	\(0.206298\pi\)
\(5\)	−8.95546	−	39.2364i	−0.160200	−	0.701883i	−0.989674	−	0.143337i	\(-0.954217\pi\)
							0.829474	−	0.558546i	\(-0.188640\pi\)
\(7\)	52.7478	+	25.4020i	0.406873	+	0.195940i	0.626115	−	0.779731i	\(-0.284644\pi\)
							−0.219242	+	0.975671i	\(0.570358\pi\)
\(11\)	−167.668	−	133.711i	−0.417801	−	0.333185i	0.391921	−	0.919999i	\(-0.371811\pi\)
							−0.809722	+	0.586814i	\(0.800382\pi\)
\(13\)	−305.252	+	382.775i	−0.500957	+	0.628181i	−0.966445	−	0.256874i	\(-0.917308\pi\)
							0.465488	+	0.885054i	\(0.345879\pi\)
\(17\)			904.119i			0.758758i	0.925241	+	0.379379i	\(0.123862\pi\)
							−0.925241	+	0.379379i	\(0.876138\pi\)
\(19\)	1184.53	+	2459.70i	0.752768	+	1.56314i	0.824596	+	0.565721i	\(0.191402\pi\)
							−0.0718286	+	0.997417i	\(0.522883\pi\)
\(23\)	−581.816	+	2549.10i	−0.229333	+	1.00477i	0.720853	+	0.693088i	\(0.243750\pi\)
							−0.950186	+	0.311684i	\(0.899107\pi\)
\(29\)	4282.58	+	1473.32i	0.945606	+	0.325313i
							\(31\)	−6513.06	+	1486.56i	−1.21725	+	0.277830i	−0.782461	−	0.622700i	\(-0.786036\pi\)
−0.434793	+	0.900530i	\(0.643178\pi\)
\(37\)	−721.625	+	575.476i	−0.0866577	+	0.0691072i	−0.665860	−	0.746076i	\(-0.731935\pi\)
							0.579203	+	0.815184i	\(0.303364\pi\)
\(41\)		−	3395.51i		−	0.315461i	−0.987482	−	0.157731i	\(-0.949582\pi\)
							0.987482	−	0.157731i	\(-0.0504177\pi\)
\(43\)	14974.2	+	3417.77i	1.23502	+	0.281884i	0.789702	−	0.613490i	\(-0.210235\pi\)
							0.445314	+	0.895374i	\(0.353092\pi\)
\(47\)	3063.73	+	2443.25i	0.202305	+	0.161333i	0.719402	−	0.694594i	\(-0.244416\pi\)
							−0.517097	+	0.855927i	\(0.672987\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.6.e.a.22.1	yes	72
29.2	odd	28		841.6.a.l.1.1		72
29.4	even	14	inner	29.6.e.a.4.1	✓	72
29.27	odd	28		841.6.a.l.1.72		72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.6.e.a.4.1	✓	72	29.4	even	14	inner
29.6.e.a.22.1	yes	72	1.1	even	1	trivial
841.6.a.l.1.1		72	29.2	odd	28
841.6.a.l.1.72		72	29.27	odd	28