Embedded newform 69.3.f.a.10.4

• Label: 69.3.f.a.10.4
• Level: $69$
• Weight: $3$
• Character: 69.10
• 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			10.4
Character	\(\chi\)	\(=\)	69.10
Dual form			69.3.f.a.7.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.141760 + 0.163600i) q^{2} +(1.45709 + 0.936417i) q^{3} +(0.562590 - 3.91290i) q^{4} +(2.81506 + 1.28559i) q^{5} +(0.0533600 + 0.371127i) q^{6} +(0.266428 - 0.907372i) q^{7} +(1.44834 - 0.930792i) q^{8} +(1.24625 + 2.72890i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q + 4 q^{2} - 12 q^{6} - 4 q^{8} - 24 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(28\)	\(47\)
\(\chi(n)\)	\(e\left(\frac{3}{22}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.141760	+	0.163600i	0.0708801	+	0.0818000i	0.790082	−	0.613001i	\(-0.210038\pi\)
							−0.719202	+	0.694801i	\(0.755492\pi\)
\(3\)	1.45709	+	0.936417i	0.485698	+	0.312139i
							\(5\)	2.81506	+	1.28559i	0.563011	+	0.257119i	0.676527	−	0.736418i	\(-0.263484\pi\)
−0.113516	+	0.993536i	\(0.536211\pi\)
\(7\)	0.266428	−	0.907372i	0.0380612	−	0.129625i	−0.938271	−	0.345901i	\(-0.887573\pi\)
							0.976332	+	0.216277i	\(0.0693914\pi\)
\(11\)	6.07766	+	5.26632i	0.552514	+	0.478756i	0.885798	−	0.464070i	\(-0.153611\pi\)
							−0.333284	+	0.942826i	\(0.608157\pi\)
\(13\)	−8.46606	+	2.48586i	−0.651236	+	0.191220i	−0.590631	−	0.806942i	\(-0.701121\pi\)
							−0.0606047	+	0.998162i	\(0.519303\pi\)
\(17\)	1.57453	−	0.226383i	0.0926195	−	0.0133167i	−0.0958493	−	0.995396i	\(-0.530557\pi\)
							0.188469	+	0.982079i	\(0.439648\pi\)
\(19\)	−17.7368	−	2.55016i	−0.933514	−	0.134219i	−0.341259	−	0.939969i	\(-0.610853\pi\)
							−0.592255	+	0.805750i	\(0.701762\pi\)
\(23\)	−22.9738	+	1.09709i	−0.998862	+	0.0476996i
							\(29\)	5.50854	+	38.3127i	0.189950	+	1.32113i	0.832132	+	0.554577i	\(0.187120\pi\)
−0.642183	+	0.766552i	\(0.721971\pi\)
\(31\)	−13.8543	+	8.90364i	−0.446914	+	0.287214i	−0.744676	−	0.667426i	\(-0.767396\pi\)
							0.297762	+	0.954640i	\(0.403760\pi\)
\(37\)	36.7919	−	16.8023i	0.994375	−	0.454116i	0.149317	−	0.988789i	\(-0.452293\pi\)
							0.845059	+	0.534673i	\(0.179565\pi\)
\(41\)	0.210101	−	0.460057i	0.00512441	−	0.0112209i	−0.907052	−	0.421018i	\(-0.861673\pi\)
							0.912177	+	0.409797i	\(0.134400\pi\)
\(43\)	20.1691	−	31.3837i	0.469049	−	0.729854i	−0.523457	−	0.852052i	\(-0.675358\pi\)
							0.992506	+	0.122198i	\(0.0389943\pi\)
\(47\)	54.8139			1.16625			0.583126	−	0.812381i	\(-0.301829\pi\)
							0.583126	+	0.812381i	\(0.301829\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.3.f.a.10.4	yes	80
3.2	odd	2		207.3.j.b.10.5		80
23.7	odd	22	inner	69.3.f.a.7.4	✓	80
69.53	even	22		207.3.j.b.145.5		80

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.3.f.a.7.4	✓	80	23.7	odd	22	inner
69.3.f.a.10.4	yes	80	1.1	even	1	trivial
207.3.j.b.10.5		80	3.2	odd	2
207.3.j.b.145.5		80	69.53	even	22