Embedded newform 69.2.g.a.5.6

• Label: 69.2.g.a.5.6
• Level: $69$
• Weight: $2$
• Character: 69.5
• Analytic conductor: $0.551$
• Analytic rank: $0$
• Dimension: $60$
• 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			5.6
Character	\(\chi\)	\(=\)	69.5
Dual form			69.2.g.a.14.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.694600 - 2.36559i) q^{2} +(-1.73202 + 0.0101732i) q^{3} +(-3.43104 - 2.20500i) q^{4} +(-0.0722440 + 0.502468i) q^{5} +(-1.17900 + 4.10432i) q^{6} +(3.47479 - 1.58688i) q^{7} +(-3.87278 + 3.35579i) q^{8} +(2.99979 - 0.0352405i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60 q - 11 q^{3} - 10 q^{4} - 14 q^{6} - 22 q^{7} - 11 q^{9}+O(q^{10}) \)

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	\(-0.572127\pi\)
							0.715817	−	0.698288i	\(-0.246054\pi\)
\(3\)	−1.73202	+	0.0101732i	−0.999983	+	0.00587352i
							\(5\)	−0.0722440	+	0.502468i	−0.0323085	+	0.224711i	−0.999578	−	0.0290397i	\(-0.990755\pi\)
0.967270	+	0.253750i	\(0.0816642\pi\)
\(7\)	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	−0.131990	−	0.991251i	\(-0.542137\pi\)
							0.424874	+	0.905253i	\(0.360319\pi\)
\(13\)	−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	−0.982382	−	0.186882i	\(-0.940162\pi\)
							−0.238102	+	0.971240i	\(0.576525\pi\)
\(19\)	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	0.798661	−	0.601781i	\(-0.205542\pi\)
−0.879175	+	0.476499i	\(0.841906\pi\)
\(31\)	−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	0.274436	−	0.961605i	\(-0.411509\pi\)
							−0.00759585	+	0.999971i	\(0.502418\pi\)
\(43\)	−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	\(-0.981842\pi\)
							0.998373	−	0.0570150i	\(-0.0181583\pi\)

(See \(a_n\) instead)

Twists

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

    

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