Embedded newform 29.6.d.a.20.10

• Label: 29.6.d.a.20.10
• Level: $29$
• Weight: $6$
• Character: 29.20
• Analytic conductor: $4.651$
• Analytic rank: $0$
• Dimension: $66$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(4.65113077458\)
Analytic rank:	\(0\)
Dimension:	\(66\)
Relative dimension:	\(11\) over \(\Q(\zeta_{7})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{7}]$

Embedding invariants

Embedding label			20.10
Character	\(\chi\)	\(=\)	29.20
Dual form			29.6.d.a.16.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(8.11483 - 3.90790i) q^{2} +(14.4847 + 18.1632i) q^{3} +(30.6271 - 38.4052i) q^{4} +(-25.2212 + 12.1459i) q^{5} +(188.520 + 90.7866i) q^{6} +(-48.0050 - 60.1963i) q^{7} +(34.3161 - 150.349i) q^{8} +(-66.0234 + 289.267i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 66 q - 11 q^{2} - 5 q^{3} - 171 q^{4} + 29 q^{5} + 445 q^{6} + 17 q^{7} + 1046 q^{8} - 732 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{6}{7}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	8.11483	−	3.90790i	1.43451	−	0.690825i	0.454682	−	0.890654i	\(-0.349753\pi\)
							0.979831	+	0.199829i	\(0.0640386\pi\)
\(3\)	14.4847	+	18.1632i	0.929191	+	1.16517i	0.985994	+	0.166783i	\(0.0533380\pi\)
							−0.0568027	+	0.998385i	\(0.518091\pi\)
\(5\)	−25.2212	+	12.1459i	−0.451171	+	0.217273i	−0.645649	−	0.763634i	\(-0.723413\pi\)
							0.194478	+	0.980907i	\(0.437699\pi\)
\(7\)	−48.0050	−	60.1963i	−0.370289	−	0.464328i	0.561421	−	0.827530i	\(-0.310255\pi\)
							−0.931710	+	0.363202i	\(0.881683\pi\)
\(11\)	−64.3008	−	281.720i	−0.160227	−	0.701998i	−0.989665	−	0.143400i	\(-0.954196\pi\)
							0.829438	−	0.558598i	\(-0.188661\pi\)
\(13\)	−233.936	−	1024.94i	−0.383918	−	1.68206i	−0.685068	−	0.728479i	\(-0.740228\pi\)
							0.301150	−	0.953577i	\(-0.402629\pi\)
\(17\)	971.108			0.814977			0.407488	−	0.913210i	\(-0.366405\pi\)
							0.407488	+	0.913210i	\(0.366405\pi\)
\(19\)	−1778.91	+	2230.68i	−1.13050	+	1.41760i	−0.235307	+	0.971921i	\(0.575609\pi\)
							−0.895192	+	0.445680i	\(0.852962\pi\)
\(23\)	2405.04	+	1158.21i	0.947989	+	0.456528i	0.842981	−	0.537944i	\(-0.180799\pi\)
							0.105009	+	0.994471i	\(0.466513\pi\)
\(29\)	−4088.22	+	1948.74i	−0.902691	+	0.430288i
							\(31\)	4467.16	−	2151.27i	0.834885	−	0.402060i	0.0329402	−	0.999457i	\(-0.489513\pi\)
0.801945	+	0.597398i	\(0.203799\pi\)
\(37\)	106.509	−	466.646i	0.0127903	−	0.0560381i	−0.968130	−	0.250448i	\(-0.919422\pi\)
							0.980920	+	0.194410i	\(0.0622792\pi\)
\(41\)	−8379.08			−0.778461			−0.389230	−	0.921140i	\(-0.627259\pi\)
							−0.389230	+	0.921140i	\(0.627259\pi\)
\(43\)	12646.6	+	6090.27i	1.04304	+	0.502303i	0.875326	−	0.483533i	\(-0.160647\pi\)
							0.167716	+	0.985835i	\(0.446361\pi\)
\(47\)	−1821.38	−	7980.00i	−0.120270	−	0.526936i	−0.998788	−	0.0492268i	\(-0.984324\pi\)
							0.878518	−	0.477709i	\(-0.158533\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.6.d.a.20.10	yes	66
29.4	even	14		841.6.a.h.1.30		33
29.16	even	7	inner	29.6.d.a.16.10	✓	66
29.25	even	7		841.6.a.i.1.4		33

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.6.d.a.16.10	✓	66	29.16	even	7	inner
29.6.d.a.20.10	yes	66	1.1	even	1	trivial
841.6.a.h.1.30		33	29.4	even	14
841.6.a.i.1.4		33	29.25	even	7