Embedded newform 69.4.g.a.5.6

• Label: 69.4.g.a.5.6
• Level: $69$
• Weight: $4$
• Character: 69.5
• Analytic conductor: $4.071$
• Analytic rank: $0$
• Dimension: $220$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 69 = 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	69.g (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.07113179040\)
Analytic rank:	\(0\)
Dimension:	\(220\)
Relative dimension:	\(22\) 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			69.4.g.a.14.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.978191 + 3.33141i) q^{2} +(-4.71623 - 2.18110i) q^{3} +(-3.41142 - 2.19239i) q^{4} +(1.70238 - 11.8403i) q^{5} +(11.8795 - 13.5782i) q^{6} +(22.3835 - 10.2222i) q^{7} +(-10.3513 + 8.96943i) q^{8} +(17.4856 + 20.5731i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 220 q - 5 q^{3} + 50 q^{4} - 32 q^{6} - 22 q^{7} - 5 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.978191	+	3.33141i	−0.345843	+	1.17783i	0.584578	+	0.811338i	\(0.301260\pi\)
							−0.930420	+	0.366494i	\(0.880558\pi\)
\(3\)	−4.71623	−	2.18110i	−0.907639	−	0.419752i
							\(5\)	1.70238	−	11.8403i	0.152266	−	1.05903i	−0.760144	−	0.649755i	\(-0.774872\pi\)
0.912410	−	0.409278i	\(-0.134219\pi\)
\(7\)	22.3835	−	10.2222i	1.20860	−	0.551947i	0.293801	−	0.955867i	\(-0.405080\pi\)
							0.914794	+	0.403920i	\(0.132352\pi\)
\(11\)	9.56696	−	2.80911i	0.262232	−	0.0769982i	−0.147974	−	0.988991i	\(-0.547275\pi\)
							0.410206	+	0.911993i	\(0.365457\pi\)
\(13\)	9.77807	−	21.4110i	0.208612	−	0.456795i	−0.776185	−	0.630505i	\(-0.782848\pi\)
							0.984797	+	0.173709i	\(0.0555753\pi\)
\(17\)	82.6502	−	53.1160i	1.17915	−	0.757796i	0.203923	−	0.978987i	\(-0.434631\pi\)
							0.975231	+	0.221191i	\(0.0709944\pi\)
\(19\)	33.7207	−	52.4704i	0.407161	−	0.633555i	−0.575752	−	0.817624i	\(-0.695291\pi\)
							0.982913	+	0.184070i	\(0.0589272\pi\)
\(23\)	61.2459	+	91.7384i	0.555246	+	0.831686i
							\(29\)	−88.5160	−	137.734i	−0.566793	−	0.881947i	0.433018	−	0.901385i	\(-0.357449\pi\)
−0.999811	+	0.0194380i	\(0.993812\pi\)
\(31\)	−98.0439	−	113.149i	−0.568039	−	0.655552i	0.396951	−	0.917840i	\(-0.370068\pi\)
							−0.964990	+	0.262288i	\(0.915523\pi\)
\(37\)	−318.164	+	45.7450i	−1.41367	+	0.203255i	−0.806461	−	0.591287i	\(-0.798620\pi\)
							−0.607209	+	0.794542i	\(0.707711\pi\)
\(41\)	9.23988	+	1.32849i	0.0351958	+	0.00506039i	0.159890	−	0.987135i	\(-0.448886\pi\)
							−0.124694	+	0.992195i	\(0.539795\pi\)
\(43\)	168.296	+	145.829i	0.596859	+	0.517181i	0.900072	−	0.435742i	\(-0.143514\pi\)
							−0.303213	+	0.952923i	\(0.598059\pi\)
\(47\)			632.882i			1.96416i	0.188477	+	0.982078i	\(0.439645\pi\)
							−0.188477	+	0.982078i	\(0.560355\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.4.g.a.5.6	✓	220
3.2	odd	2	inner	69.4.g.a.5.17	yes	220
23.14	odd	22	inner	69.4.g.a.14.17	yes	220
69.14	even	22	inner	69.4.g.a.14.6	yes	220

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.4.g.a.5.6	✓	220	1.1	even	1	trivial
69.4.g.a.5.17	yes	220	3.2	odd	2	inner
69.4.g.a.14.6	yes	220	69.14	even	22	inner
69.4.g.a.14.17	yes	220	23.14	odd	22	inner