Embedded newform 29.6.e.a.13.5

• Label: 29.6.e.a.13.5
• 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			13.5
Character	\(\chi\)	\(=\)	29.13
Dual form			29.6.e.a.9.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.70194 + 3.53411i) q^{2} +(-5.72286 - 4.56383i) q^{3} +(10.3583 + 12.9889i) q^{4} +(-74.2917 - 35.7770i) q^{5} +(25.8690 - 12.4579i) q^{6} +(68.1445 - 85.4504i) q^{7} +(-185.909 + 42.4324i) q^{8} +(-42.1500 - 184.671i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 7 q^{2} - 7 q^{3} + 165 q^{4} - 53 q^{5} - 311 q^{6} - 27 q^{7} - 1498 q^{8} + 1567 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.70194	+	3.53411i	−0.300863	+	0.624748i	−0.995516	−	0.0945970i	\(-0.969844\pi\)
							0.694653	+	0.719345i	\(0.255558\pi\)
\(3\)	−5.72286	−	4.56383i	−0.367121	−	0.292770i	0.422501	−	0.906362i	\(-0.361152\pi\)
							−0.789623	+	0.613593i	\(0.789724\pi\)
\(5\)	−74.2917	−	35.7770i	−1.32897	−	0.639998i	−0.371474	−	0.928443i	\(-0.621147\pi\)
							−0.957496	+	0.288445i	\(0.906862\pi\)
\(7\)	68.1445	−	85.4504i	0.525636	−	0.659127i	−0.446159	−	0.894954i	\(-0.647208\pi\)
							0.971795	+	0.235827i	\(0.0757798\pi\)
\(11\)	−364.340	−	83.1581i	−0.907872	−	0.207216i	−0.257007	−	0.966410i	\(-0.582736\pi\)
							−0.650865	+	0.759194i	\(0.725594\pi\)
\(13\)	171.325	−	750.625i	0.281166	−	1.23187i	−0.615134	−	0.788422i	\(-0.710898\pi\)
							0.896300	−	0.443447i	\(-0.146245\pi\)
\(17\)			1720.54i			1.44392i	0.691935	+	0.721960i	\(0.256758\pi\)
							−0.691935	+	0.721960i	\(0.743242\pi\)
\(19\)	−1706.04	+	1360.52i	−1.08419	+	0.864613i	−0.991373	−	0.131070i	\(-0.958159\pi\)
							−0.0928170	+	0.995683i	\(0.529587\pi\)
\(23\)	−1074.98	+	517.681i	−0.423720	+	0.204053i	−0.633578	−	0.773679i	\(-0.718414\pi\)
							0.209858	+	0.977732i	\(0.432700\pi\)
\(29\)	2659.42	−	3665.87i	0.587208	−	0.809436i
							\(31\)	4056.62	−	8423.66i	0.758158	−	1.57433i	−0.0592294	−	0.998244i	\(-0.518864\pi\)
0.817388	−	0.576088i	\(-0.195421\pi\)
\(37\)	−7360.98	+	1680.10i	−0.883958	+	0.201758i	−0.640320	−	0.768109i	\(-0.721198\pi\)
							−0.243638	+	0.969866i	\(0.578341\pi\)
\(41\)		−	6963.40i		−	0.646937i	−0.946239	−	0.323468i	\(-0.895151\pi\)
							0.946239	−	0.323468i	\(-0.104849\pi\)
\(43\)	2068.01	+	4294.27i	0.170562	+	0.354175i	0.968675	−	0.248332i	\(-0.0798825\pi\)
							−0.798113	+	0.602508i	\(0.794168\pi\)
\(47\)	10329.7	+	2357.68i	0.682090	+	0.155683i	0.549507	−	0.835489i	\(-0.314816\pi\)
							0.132583	+	0.991172i	\(0.457673\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.6.e.a.13.5	yes	72
29.3	odd	28		841.6.a.l.1.26		72
29.9	even	14	inner	29.6.e.a.9.5	✓	72
29.26	odd	28		841.6.a.l.1.47		72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.6.e.a.9.5	✓	72	29.9	even	14	inner
29.6.e.a.13.5	yes	72	1.1	even	1	trivial
841.6.a.l.1.26		72	29.3	odd	28
841.6.a.l.1.47		72	29.26	odd	28