Embedded newform 69.3.f.a.7.3

• Label: 69.3.f.a.7.3
• Level: $69$
• Weight: $3$
• Character: 69.7
• Analytic conductor: $1.880$
• Analytic rank: $0$
• Dimension: $80$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(1.88011382409\)
Analytic rank:	\(0\)
Dimension:	\(80\)
Relative dimension:	\(8\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			7.3
Character	\(\chi\)	\(=\)	69.7
Dual form			69.3.f.a.10.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.71450 + 1.97864i) q^{2} +(1.45709 - 0.936417i) q^{3} +(-0.406240 - 2.82546i) q^{4} +(-6.24106 + 2.85020i) q^{5} +(-0.645356 + 4.48855i) q^{6} +(3.18154 + 10.8353i) q^{7} +(-2.52293 - 1.62139i) q^{8} +(1.24625 - 2.72890i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q + 4 q^{2} - 12 q^{6} - 4 q^{8} - 24 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(28\)	\(47\)
\(\chi(n)\)	\(e\left(\frac{19}{22}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.71450	+	1.97864i	−0.857250	+	0.989319i	−1.00000		0.000440029i	\(-0.999860\pi\)
							0.142750	+	0.989759i	\(0.454405\pi\)
\(3\)	1.45709	−	0.936417i	0.485698	−	0.312139i
							\(5\)	−6.24106	+	2.85020i	−1.24821	+	0.570039i	−0.926320	−	0.376737i	\(-0.877046\pi\)
−0.321892	+	0.946776i	\(0.604319\pi\)
\(7\)	3.18154	+	10.8353i	0.454506	+	1.54790i	0.794373	+	0.607430i	\(0.207800\pi\)
							−0.339867	+	0.940473i	\(0.610382\pi\)
\(11\)	−7.10492	+	6.15645i	−0.645902	+	0.559677i	−0.915009	−	0.403433i	\(-0.867817\pi\)
							0.269107	+	0.963110i	\(0.413271\pi\)
\(13\)	−11.7602	−	3.45310i	−0.904629	−	0.265623i	−0.203851	−	0.979002i	\(-0.565346\pi\)
							−0.700779	+	0.713379i	\(0.747164\pi\)
\(17\)	31.6725	+	4.55382i	1.86309	+	0.267872i	0.979656	−	0.200683i	\(-0.0643160\pi\)
							0.883434	+	0.468555i	\(0.155225\pi\)
\(19\)	−3.50992	+	0.504650i	−0.184733	+	0.0265605i	−0.234060	−	0.972222i	\(-0.575201\pi\)
							0.0493275	+	0.998783i	\(0.484292\pi\)
\(23\)	19.1735	+	12.7034i	0.833630	+	0.552323i
							\(29\)	−0.469485	+	3.26534i	−0.0161891	+	0.112598i	−0.996314	−	0.0857855i	\(-0.972660\pi\)
0.980124	+	0.198383i	\(0.0635691\pi\)
\(31\)	23.0764	+	14.8303i	0.744400	+	0.478397i	0.857047	−	0.515238i	\(-0.172297\pi\)
							−0.112647	+	0.993635i	\(0.535933\pi\)
\(37\)	9.63496	+	4.40014i	0.260404	+	0.118923i	0.541340	−	0.840804i	\(-0.317917\pi\)
							−0.280935	+	0.959727i	\(0.590645\pi\)
\(41\)	−10.0767	−	22.0649i	−0.245774	−	0.538169i	0.746034	−	0.665907i	\(-0.231955\pi\)
							−0.991808	+	0.127739i	\(0.959228\pi\)
\(43\)	−29.7425	−	46.2802i	−0.691686	−	1.07628i	−0.992458	−	0.122588i	\(-0.960881\pi\)
							0.300772	−	0.953696i	\(-0.402756\pi\)
\(47\)	43.5310			0.926192			0.463096	−	0.886308i	\(-0.346738\pi\)
							0.463096	+	0.886308i	\(0.346738\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.3.f.a.7.3	✓	80
3.2	odd	2		207.3.j.b.145.6		80
23.10	odd	22	inner	69.3.f.a.10.3	yes	80
69.56	even	22		207.3.j.b.10.6		80

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.3.f.a.7.3	✓	80	1.1	even	1	trivial
69.3.f.a.10.3	yes	80	23.10	odd	22	inner
207.3.j.b.10.6		80	69.56	even	22
207.3.j.b.145.6		80	3.2	odd	2