Embedded newform 69.6.g.a.5.20

• Label: 69.6.g.a.5.20
• Level: $69$
• Weight: $6$
• Character: 69.5
• Analytic conductor: $11.066$
• Analytic rank: $0$
• Dimension: $380$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 69 = 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	69.g (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.0664835671\)
Analytic rank:	\(0\)
Dimension:	\(380\)
Relative dimension:	\(38\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			5.20
Character	\(\chi\)	\(=\)	69.5
Dual form			69.6.g.a.14.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.190214 - 0.647810i) q^{2} +(-0.946612 + 15.5597i) q^{3} +(26.5366 + 17.0541i) q^{4} +(6.91235 - 48.0764i) q^{5} +(9.89966 + 3.57290i) q^{6} +(156.619 - 71.5256i) q^{7} +(32.4235 - 28.0951i) q^{8} +(-241.208 - 29.4580i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 380 q - 11 q^{3} + 578 q^{4} + 64 q^{6} - 22 q^{7} + 433 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/69\mathbb{Z}\right)^\times\).

\(n\)	\(28\)	\(47\)
\(\chi(n)\)	\(e\left(\frac{1}{22}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.190214	−	0.647810i	0.0336254	−	0.114518i	−0.940970	−	0.338489i	\(-0.890084\pi\)
							0.974596	+	0.223972i	\(0.0719023\pi\)
\(3\)	−0.946612	+	15.5597i	−0.0607252	+	0.998155i
							\(5\)	6.91235	−	48.0764i	0.123652	−	0.860017i	−0.829712	−	0.558192i	\(-0.811495\pi\)
0.953363	−	0.301825i	\(-0.0975958\pi\)
\(7\)	156.619	−	71.5256i	1.20809	−	0.551717i	0.293447	−	0.955975i	\(-0.405198\pi\)
							0.914645	+	0.404259i	\(0.132470\pi\)
\(11\)	600.835	−	176.421i	1.49718	−	0.439611i	0.572355	−	0.820006i	\(-0.306030\pi\)
							0.924824	+	0.380394i	\(0.124212\pi\)
\(13\)	−230.379	+	504.461i	−0.378081	+	0.827883i	0.620949	+	0.783851i	\(0.286748\pi\)
							−0.999030	+	0.0440316i	\(0.985980\pi\)
\(17\)	−259.023	+	166.464i	−0.217378	+	0.139700i	−0.644799	−	0.764352i	\(-0.723059\pi\)
							0.427420	+	0.904053i	\(0.359422\pi\)
\(19\)	−828.997	+	1289.94i	−0.526828	+	0.819761i	−0.998061	−	0.0622485i	\(-0.980173\pi\)
							0.471233	+	0.882009i	\(0.343809\pi\)
\(23\)	2044.96	−	1501.50i	0.806055	−	0.591841i
							\(29\)	2625.45	+	4085.28i	0.579707	+	0.902041i	0.999986	−	0.00535364i	\(-0.00170412\pi\)
−0.420279	+	0.907395i	\(0.638068\pi\)
\(31\)	−4374.58	−	5048.53i	−0.817583	−	0.943541i	0.181624	−	0.983368i	\(-0.441865\pi\)
							−0.999207	+	0.0398272i	\(0.987319\pi\)
\(37\)	−10053.0	+	1445.40i	−1.20723	+	0.173574i	−0.716406	−	0.697683i	\(-0.754214\pi\)
							−0.490827	+	0.871257i	\(0.663305\pi\)
\(41\)	−12174.1	−	1750.37i	−1.13104	−	0.162618i	−0.448717	−	0.893674i	\(-0.648119\pi\)
							−0.682318	+	0.731055i	\(0.739028\pi\)
\(43\)	13024.2	+	11285.5i	1.07419	+	0.930789i	0.997800	−	0.0662895i	\(-0.0211161\pi\)
							0.0763870	+	0.997078i	\(0.475662\pi\)
\(47\)			2734.58i			0.180570i	0.995916	+	0.0902850i	\(0.0287778\pi\)
							−0.995916	+	0.0902850i	\(0.971222\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.6.g.a.5.20	yes	380
3.2	odd	2	inner	69.6.g.a.5.19	✓	380
23.14	odd	22	inner	69.6.g.a.14.19	yes	380
69.14	even	22	inner	69.6.g.a.14.20	yes	380

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.6.g.a.5.19	✓	380	3.2	odd	2	inner
69.6.g.a.5.20	yes	380	1.1	even	1	trivial
69.6.g.a.14.19	yes	380	23.14	odd	22	inner
69.6.g.a.14.20	yes	380	69.14	even	22	inner