Embedded newform 69.3.f.a.10.8

• Label: 69.3.f.a.10.8
• 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			10.8
Character	\(\chi\)	\(=\)	69.10
Dual form			69.3.f.a.7.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.55609 + 2.94989i) q^{2} +(-1.45709 - 0.936417i) q^{3} +(-1.59897 + 11.1211i) q^{4} +(1.23125 + 0.562293i) q^{5} +(-0.962140 - 6.69183i) q^{6} +(3.18330 - 10.8413i) q^{7} +(-23.7585 + 15.2686i) q^{8} +(1.24625 + 2.72890i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q + 4 q^{2} - 12 q^{6} - 4 q^{8} - 24 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.55609	+	2.94989i	1.27805	+	1.47494i	0.804229	+	0.594319i	\(0.202579\pi\)
							0.473816	+	0.880624i	\(0.342876\pi\)
\(3\)	−1.45709	−	0.936417i	−0.485698	−	0.312139i
							\(5\)	1.23125	+	0.562293i	0.246250	+	0.112459i	0.534717	−	0.845031i	\(-0.320418\pi\)
−0.288467	+	0.957490i	\(0.593146\pi\)
\(7\)	3.18330	−	10.8413i	0.454757	−	1.54876i	−0.339156	−	0.940730i	\(-0.610141\pi\)
							0.793913	−	0.608031i	\(-0.208040\pi\)
\(11\)	2.14628	+	1.85977i	0.195117	+	0.169070i	0.746940	−	0.664891i	\(-0.231522\pi\)
							−0.551823	+	0.833961i	\(0.686068\pi\)
\(13\)	5.86393	−	1.72181i	0.451072	−	0.132447i	−0.0483030	−	0.998833i	\(-0.515381\pi\)
							0.499375	+	0.866386i	\(0.333563\pi\)
\(17\)	16.0817	−	2.31221i	0.945985	−	0.136012i	0.347978	−	0.937503i	\(-0.386869\pi\)
							0.598007	+	0.801491i	\(0.295959\pi\)
\(19\)	−11.0740	−	1.59220i	−0.582842	−	0.0838000i	−0.155414	−	0.987849i	\(-0.549671\pi\)
							−0.427428	+	0.904049i	\(0.640580\pi\)
\(23\)	−8.39169	−	21.4145i	−0.364856	−	0.931064i
							\(29\)	4.32037	+	30.0489i	0.148978	+	1.03617i	0.917896	+	0.396820i	\(0.129886\pi\)
−0.768918	+	0.639347i	\(0.779205\pi\)
\(31\)	−20.1715	+	12.9634i	−0.650692	+	0.418174i	−0.823919	−	0.566707i	\(-0.808217\pi\)
							0.173227	+	0.984882i	\(0.444581\pi\)
\(37\)	−52.4022	+	23.9313i	−1.41627	+	0.646791i	−0.968877	−	0.247544i	\(-0.920377\pi\)
							−0.447398	+	0.894335i	\(0.647649\pi\)
\(41\)	−14.2803	+	31.2695i	−0.348300	+	0.762670i	0.651691	+	0.758484i	\(0.274060\pi\)
							−0.999991	+	0.00418607i	\(0.998668\pi\)
\(43\)	10.3117	−	16.0453i	0.239807	−	0.373147i	−0.700400	−	0.713750i	\(-0.746995\pi\)
							0.940207	+	0.340604i	\(0.110631\pi\)
\(47\)	30.7004			0.653200			0.326600	−	0.945163i	\(-0.394097\pi\)
							0.326600	+	0.945163i	\(0.394097\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.3.f.a.10.8	yes	80
3.2	odd	2		207.3.j.b.10.1		80
23.7	odd	22	inner	69.3.f.a.7.8	✓	80
69.53	even	22		207.3.j.b.145.1		80

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.3.f.a.7.8	✓	80	23.7	odd	22	inner
69.3.f.a.10.8	yes	80	1.1	even	1	trivial
207.3.j.b.10.1		80	3.2	odd	2
207.3.j.b.145.1		80	69.53	even	22