Embedded newform 69.2.g.a.5.1

• Label: 69.2.g.a.5.1
• 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			5.1
Character	\(\chi\)	\(=\)	69.5
Dual form			69.2.g.a.14.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.694600 + 2.36559i) q^{2} +(0.236423 + 1.71584i) q^{3} +(-3.43104 - 2.20500i) q^{4} +(0.0722440 - 0.502468i) q^{5} +(-4.22319 - 0.632543i) q^{6} +(3.47479 - 1.58688i) q^{7} +(3.87278 - 3.35579i) q^{8} +(-2.88821 + 0.811326i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60 q - 11 q^{3} - 10 q^{4} - 14 q^{6} - 22 q^{7} - 11 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.694600	+	2.36559i	−0.491156	+	1.67273i	0.224661	+	0.974437i	\(0.427873\pi\)
							−0.715817	+	0.698288i	\(0.753946\pi\)
\(3\)	0.236423	+	1.71584i	0.136499	+	0.990640i
							\(5\)	0.0722440	−	0.502468i	0.0323085	−	0.224711i	−0.967270	−	0.253750i	\(-0.918336\pi\)
0.999578	+	0.0290397i	\(0.00924492\pi\)
\(7\)	3.47479	−	1.58688i	1.31335	−	0.599786i	0.369213	−	0.929345i	\(-0.379627\pi\)
							0.944135	+	0.329559i	\(0.106900\pi\)
\(11\)	−0.971385	+	0.285224i	−0.292884	+	0.0859984i	−0.424874	−	0.905253i	\(-0.639681\pi\)
							0.131990	+	0.991251i	\(0.457863\pi\)
\(13\)	−1.96293	+	4.29821i	−0.544419	+	1.19211i	0.414921	+	0.909857i	\(0.363809\pi\)
							−0.959340	+	0.282253i	\(0.908918\pi\)
\(17\)	5.03219	−	3.23399i	1.22048	−	0.784358i	0.238102	−	0.971240i	\(-0.423475\pi\)
							0.982382	+	0.186882i	\(0.0598384\pi\)
\(19\)	1.15430	−	1.79612i	0.264814	−	0.412058i	−0.683228	−	0.730205i	\(-0.739424\pi\)
							0.948041	+	0.318147i	\(0.103061\pi\)
\(23\)	−3.68290	−	3.07184i	−0.767938	−	0.640524i
							\(29\)	0.433582	+	0.674667i	0.0805142	+	0.125283i	0.879175	−	0.476499i	\(-0.158094\pi\)
−0.798661	+	0.601781i	\(0.794458\pi\)
\(31\)	−2.11763	−	2.44388i	−0.380338	−	0.438933i	0.533013	−	0.846107i	\(-0.321060\pi\)
							−0.913351	+	0.407174i	\(0.866514\pi\)
\(37\)	−4.43544	+	0.637721i	−0.729182	+	0.104841i	−0.496900	−	0.867808i	\(-0.665528\pi\)
							−0.232282	+	0.972648i	\(0.574619\pi\)
\(41\)	−1.70861	−	0.245661i	−0.266840	−	0.0383659i	0.00759585	−	0.999971i	\(-0.497582\pi\)
							−0.274436	+	0.961605i	\(0.588491\pi\)
\(43\)	−4.77133	−	4.13438i	−0.727621	−	0.630487i	0.210179	−	0.977663i	\(-0.432595\pi\)
							−0.937800	+	0.347176i	\(0.887141\pi\)
\(47\)			0.781750i			0.114030i	0.998373	+	0.0570150i	\(0.0181583\pi\)
							−0.998373	+	0.0570150i	\(0.981842\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.2.g.a.5.1	✓	60
3.2	odd	2	inner	69.2.g.a.5.6	yes	60
23.14	odd	22	inner	69.2.g.a.14.6	yes	60
69.14	even	22	inner	69.2.g.a.14.1	yes	60

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.2.g.a.5.1	✓	60	1.1	even	1	trivial
69.2.g.a.5.6	yes	60	3.2	odd	2	inner
69.2.g.a.14.1	yes	60	69.14	even	22	inner
69.2.g.a.14.6	yes	60	23.14	odd	22	inner