Embedded newform 29.8.e.a.5.1

• Label: 29.8.e.a.5.1
• Level: $29$
• Weight: $8$
• Character: 29.5
• 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			5.1
Character	\(\chi\)	\(=\)	29.5
Dual form			29.8.e.a.6.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-16.8298 + 13.4213i) q^{2} +(-72.7388 + 16.6021i) q^{3} +(74.6272 - 326.963i) q^{4} +(216.453 + 271.423i) q^{5} +(1001.35 - 1255.66i) q^{6} +(-204.467 - 895.829i) q^{7} +(1936.81 + 4021.83i) q^{8} +(3044.88 - 1466.34i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q - 7 q^{2} - 7 q^{3} + 709 q^{4} + 191 q^{5} + 1801 q^{6} - 667 q^{7} + 9086 q^{8} + 9103 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{11}{14}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−16.8298	+	13.4213i	−1.48755	+	1.18629i	−0.551628	+	0.834090i	\(0.685993\pi\)
							−0.935927	+	0.352195i	\(0.885435\pi\)
\(3\)	−72.7388	+	16.6021i	−1.55540	+	0.355009i	−0.911892	−	0.410431i	\(-0.865378\pi\)
							−0.643506	+	0.765441i	\(0.722521\pi\)
\(5\)	216.453	+	271.423i	0.774406	+	0.971074i	0.999995	−	0.00314840i	\(-0.00100217\pi\)
							−0.225589	+	0.974222i	\(0.572431\pi\)
\(7\)	−204.467	−	895.829i	−0.225310	−	0.987148i	−0.953410	−	0.301677i	\(-0.902454\pi\)
							0.728100	−	0.685471i	\(-0.240404\pi\)
\(11\)	1238.15	−	2571.05i	0.280478	−	0.582419i	−0.712370	−	0.701805i	\(-0.752378\pi\)
							0.992848	+	0.119385i	\(0.0380923\pi\)
\(13\)	−7846.50	−	3778.68i	−0.990545	−	0.477021i	−0.132826	−	0.991139i	\(-0.542405\pi\)
							−0.857719	+	0.514118i	\(0.828119\pi\)
\(17\)			11307.8i			0.558223i	0.960259	+	0.279111i	\(0.0900399\pi\)
							−0.960259	+	0.279111i	\(0.909960\pi\)
\(19\)	49323.4	+	11257.7i	1.64974	+	0.376542i	0.943485	−	0.331416i	\(-0.107526\pi\)
							0.706254	+	0.707958i	\(0.250384\pi\)
\(23\)	−5349.62	+	6708.21i	−0.0916802	+	0.114963i	−0.825556	−	0.564321i	\(-0.809138\pi\)
							0.733875	+	0.679284i	\(0.237710\pi\)
\(29\)	−131176.	−	6537.36i	−0.998760	−	0.0497748i
							\(31\)	48382.2	−	38583.5i	0.291689	−	0.232614i	−0.466701	−	0.884415i	\(-0.654558\pi\)
0.758390	+	0.651801i	\(0.225986\pi\)
\(37\)	182183.	+	378307.i	0.591291	+	1.22783i	0.955080	+	0.296348i	\(0.0957687\pi\)
							−0.363789	+	0.931481i	\(0.618517\pi\)
\(41\)			147519.i			0.334275i	0.985934	+	0.167137i	\(0.0534523\pi\)
							−0.985934	+	0.167137i	\(0.946548\pi\)
\(43\)	−227171.	−	181163.i	−0.435725	−	0.347479i	0.380931	−	0.924603i	\(-0.375603\pi\)
							−0.816657	+	0.577124i	\(0.804175\pi\)
\(47\)	−133467.	+	277147.i	−0.187513	+	0.389374i	−0.973439	−	0.228946i	\(-0.926472\pi\)
							0.785926	+	0.618320i	\(0.212186\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.8.e.a.5.1	✓	96
29.6	even	14	inner	29.8.e.a.6.1	yes	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.8.e.a.5.1	✓	96	1.1	even	1	trivial
29.8.e.a.6.1	yes	96	29.6	even	14	inner