Embedded newform 29.8.d.a.16.7

• Label: 29.8.d.a.16.7
• Level: $29$
• Weight: $8$
• Character: 29.16
• Analytic conductor: $9.059$
• Analytic rank: $0$
• Dimension: $102$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 29 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	29.d (of order \(7\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.05916573904\)
Analytic rank:	\(0\)
Dimension:	\(102\)
Relative dimension:	\(17\) over \(\Q(\zeta_{7})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{7}]$

Embedding invariants

Embedding label			16.7
Character	\(\chi\)	\(=\)	29.16
Dual form			29.8.d.a.20.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-7.56021 - 3.64081i) q^{2} +(-41.5692 + 52.1262i) q^{3} +(-35.9054 - 45.0239i) q^{4} +(-468.001 - 225.378i) q^{5} +(504.054 - 242.739i) q^{6} +(106.579 - 133.646i) q^{7} +(346.533 + 1518.26i) q^{8} +(-502.483 - 2201.52i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 102 q + q^{2} - 5 q^{3} - 1275 q^{4} + 133 q^{5} - 2675 q^{6} + 657 q^{7} - 6562 q^{8} - 15054 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{7}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−7.56021	−	3.64081i	−0.668235	−	0.321805i	0.0688287	−	0.997628i	\(-0.478074\pi\)
							−0.737063	+	0.675824i	\(0.763788\pi\)
\(3\)	−41.5692	+	52.1262i	−0.888889	+	1.11463i	0.103880	+	0.994590i	\(0.466874\pi\)
							−0.992769	+	0.120042i	\(0.961697\pi\)
\(5\)	−468.001	−	225.378i	−1.67437	−	0.806335i	−0.997533	−	0.0702051i	\(-0.977635\pi\)
							−0.676840	−	0.736130i	\(-0.736651\pi\)
\(7\)	106.579	−	133.646i	0.117443	−	0.147269i	−0.719635	−	0.694353i	\(-0.755691\pi\)
							0.837078	+	0.547084i	\(0.184262\pi\)
\(11\)	−1046.90	+	4586.76i	−0.237154	+	1.03904i	0.706398	+	0.707815i	\(0.250319\pi\)
							−0.943552	+	0.331225i	\(0.892538\pi\)
\(13\)	571.023	−	2501.81i	0.0720861	−	0.315830i	−0.926011	−	0.377497i	\(-0.876785\pi\)
							0.998097	+	0.0616675i	\(0.0196418\pi\)
\(17\)	25223.2			1.24517			0.622586	−	0.782551i	\(-0.286082\pi\)
							0.622586	+	0.782551i	\(0.286082\pi\)
\(19\)	−33934.8	−	42552.9i	−1.13503	−	1.42328i	−0.891284	−	0.453445i	\(-0.850195\pi\)
							−0.243747	−	0.969839i	\(-0.578377\pi\)
\(23\)	−24322.4	+	11713.0i	−0.416830	+	0.200735i	−0.630530	−	0.776165i	\(-0.717163\pi\)
							0.213701	+	0.976899i	\(0.431448\pi\)
\(29\)	112707.	+	67430.9i	0.858142	+	0.513412i
							\(31\)	−146516.	−	70558.5i	−0.883323	−	0.425386i	−0.0634861	−	0.997983i	\(-0.520222\pi\)
−0.819837	+	0.572597i	\(0.805936\pi\)
\(37\)	14671.1	+	64278.1i	0.0476163	+	0.208621i	0.993140	−	0.116933i	\(-0.0373064\pi\)
							−0.945523	+	0.325554i	\(0.894449\pi\)
\(41\)	453744.			1.02818			0.514088	−	0.857738i	\(-0.328131\pi\)
							0.514088	+	0.857738i	\(0.328131\pi\)
\(43\)	181502.	−	87406.7i	0.348130	−	0.167651i	−0.251644	−	0.967820i	\(-0.580971\pi\)
							0.599774	+	0.800169i	\(0.295257\pi\)
\(47\)	17541.0	−	76852.2i	0.0246440	−	0.107973i	−0.961110	−	0.276166i	\(-0.910936\pi\)
							0.985754	+	0.168193i	\(0.0537932\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.8.d.a.16.7	✓	102
29.20	even	7	inner	29.8.d.a.20.7	yes	102

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.8.d.a.16.7	✓	102	1.1	even	1	trivial
29.8.d.a.20.7	yes	102	29.20	even	7	inner