Embedded newform 69.2.g.a.5.2

• Label: 69.2.g.a.5.2
• Level: $69$
• Weight: $2$
• Character: 69.5
• Analytic conductor: $0.551$
• Analytic rank: $0$
• Dimension: $60$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 69 = 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	69.g (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.550967773947\)
Analytic rank:	\(0\)
Dimension:	\(60\)
Relative dimension:	\(6\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			5.2
Character	\(\chi\)	\(=\)	69.5
Dual form			69.2.g.a.14.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.417491 + 1.42184i) q^{2} +(1.61954 - 0.614080i) q^{3} +(-0.164830 - 0.105930i) q^{4} +(0.394240 - 2.74200i) q^{5} +(0.196983 + 2.55910i) q^{6} +(-3.84519 + 1.75604i) q^{7} +(-2.02041 + 1.75070i) q^{8} +(2.24581 - 1.98905i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60 q - 11 q^{3} - 10 q^{4} - 14 q^{6} - 22 q^{7} - 11 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.417491	+	1.42184i	−0.295210	+	1.00539i	0.669660	+	0.742668i	\(0.266440\pi\)
							−0.964870	+	0.262726i	\(0.915378\pi\)
\(3\)	1.61954	−	0.614080i	0.935041	−	0.354539i
							\(5\)	0.394240	−	2.74200i	0.176309	−	1.22626i	−0.688903	−	0.724853i	\(-0.741908\pi\)
0.865213	−	0.501405i	\(-0.167183\pi\)
\(7\)	−3.84519	+	1.75604i	−1.45335	+	0.663721i	−0.976549	−	0.215293i	\(-0.930929\pi\)
							−0.476796	+	0.879014i	\(0.658202\pi\)
\(11\)	−2.83853	+	0.833469i	−0.855850	+	0.251300i	−0.680086	−	0.733132i	\(-0.738058\pi\)
							−0.175764	+	0.984432i	\(0.556240\pi\)
\(13\)	−0.728021	+	1.59414i	−0.201917	+	0.442136i	−0.983318	−	0.181892i	\(-0.941778\pi\)
							0.781402	+	0.624028i	\(0.214505\pi\)
\(17\)	2.46194	−	1.58219i	0.597108	−	0.383738i	−0.206895	−	0.978363i	\(-0.566336\pi\)
							0.804003	+	0.594625i	\(0.202700\pi\)
\(19\)	2.28200	−	3.55086i	0.523526	−	0.814623i	−0.474311	−	0.880357i	\(-0.657303\pi\)
							0.997837	+	0.0657346i	\(0.0209391\pi\)
\(23\)	1.93001	+	4.39034i	0.402434	+	0.915449i
							\(29\)	−1.05713	−	1.64493i	−0.196304	−	0.305455i	0.729122	−	0.684384i	\(-0.239929\pi\)
−0.925426	+	0.378929i	\(0.876292\pi\)
\(31\)	2.77045	+	3.19727i	0.497587	+	0.574246i	0.947877	−	0.318636i	\(-0.103225\pi\)
							−0.450290	+	0.892882i	\(0.648679\pi\)
\(37\)	−1.82024	+	0.261710i	−0.299245	+	0.0430249i	−0.290302	−	0.956935i	\(-0.593756\pi\)
							−0.00894295	+	0.999960i	\(0.502847\pi\)
\(41\)	5.70336	+	0.820019i	0.890715	+	0.128065i	0.572453	−	0.819937i	\(-0.305992\pi\)
							0.318262	+	0.948003i	\(0.396901\pi\)
\(43\)	−3.16627	−	2.74359i	−0.482852	−	0.418394i	0.379122	−	0.925347i	\(-0.376226\pi\)
							−0.861974	+	0.506953i	\(0.830772\pi\)
\(47\)			1.46762i			0.214075i	0.994255	+	0.107037i	\(0.0341364\pi\)
							−0.994255	+	0.107037i	\(0.965864\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.2.g.a.5.2	✓	60
3.2	odd	2	inner	69.2.g.a.5.5	yes	60
23.14	odd	22	inner	69.2.g.a.14.5	yes	60
69.14	even	22	inner	69.2.g.a.14.2	yes	60

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.2.g.a.5.2	✓	60	1.1	even	1	trivial
69.2.g.a.5.5	yes	60	3.2	odd	2	inner
69.2.g.a.14.2	yes	60	69.14	even	22	inner
69.2.g.a.14.5	yes	60	23.14	odd	22	inner