Embedded newform 69.2.g.a.11.3

• Label: 69.2.g.a.11.3
• Level: $69$
• Weight: $2$
• Character: 69.11
• 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			11.3
Character	\(\chi\)	\(=\)	69.11
Dual form			69.2.g.a.44.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0493131 - 0.0767326i) q^{2} +(1.73042 - 0.0750729i) q^{3} +(0.827374 - 1.81170i) q^{4} +(-2.86859 + 0.842294i) q^{5} +(-0.0910930 - 0.129078i) q^{6} +(-1.75762 + 1.52299i) q^{7} +(-0.360384 + 0.0518154i) q^{8} +(2.98873 - 0.259816i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60 q - 11 q^{3} - 10 q^{4} - 14 q^{6} - 22 q^{7} - 11 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/69\mathbb{Z}\right)^\times\).

\(n\)	\(28\)	\(47\)
\(\chi(n)\)	\(e\left(\frac{9}{22}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0493131	−	0.0767326i	−0.0348696	−	0.0542581i	0.823381	−	0.567489i	\(-0.192085\pi\)
							−0.858251	+	0.513231i	\(0.828449\pi\)
\(3\)	1.73042	−	0.0750729i	0.999060	−	0.0433433i
							\(5\)	−2.86859	+	0.842294i	−1.28287	+	0.376685i	−0.850960	−	0.525230i	\(-0.823979\pi\)
−0.431913	+	0.901915i	\(0.642161\pi\)
\(7\)	−1.75762	+	1.52299i	−0.664318	+	0.575635i	−0.920380	−	0.391025i	\(-0.872120\pi\)
							0.256062	+	0.966660i	\(0.417575\pi\)
\(11\)	1.83549	+	1.17959i	0.553420	+	0.355661i	0.787267	−	0.616612i	\(-0.211495\pi\)
							−0.233847	+	0.972273i	\(0.575132\pi\)
\(13\)	−2.74970	+	3.17332i	−0.762629	+	0.880120i	−0.995728	−	0.0923325i	\(-0.970568\pi\)
							0.233100	+	0.972453i	\(0.425113\pi\)
\(17\)	−1.67699	−	3.67210i	−0.406731	−	0.890616i	−0.996543	−	0.0830757i	\(-0.973526\pi\)
							0.589813	−	0.807540i	\(-0.299202\pi\)
\(19\)	2.40861	+	1.09998i	0.552574	+	0.252352i	0.672076	−	0.740482i	\(-0.265403\pi\)
							−0.119502	+	0.992834i	\(0.538130\pi\)
\(23\)	−3.72801	−	3.01694i	−0.777344	−	0.629076i
							\(29\)	5.44962	−	2.48876i	1.01197	−	0.462151i	0.160769	−	0.986992i	\(-0.448603\pi\)
0.851200	+	0.524841i	\(0.175875\pi\)
\(31\)	−0.113954	−	0.792566i	−0.0204667	−	0.142349i	0.977026	−	0.213121i	\(-0.0683629\pi\)
							−0.997493	+	0.0707719i	\(0.977454\pi\)
\(37\)	2.32136	−	7.90582i	0.381629	−	1.29971i	−0.515101	−	0.857130i	\(-0.672245\pi\)
							0.896729	−	0.442579i	\(-0.145936\pi\)
\(41\)	2.47018	+	8.41266i	0.385777	+	1.31384i	0.892235	+	0.451572i	\(0.149136\pi\)
							−0.506458	+	0.862265i	\(0.669045\pi\)
\(43\)	−2.82529	−	0.406215i	−0.430853	−	0.0619473i	−0.0765230	−	0.997068i	\(-0.524382\pi\)
							−0.354330	+	0.935121i	\(0.615291\pi\)
\(47\)		−	1.51008i		−	0.220268i	−0.993917	−	0.110134i	\(-0.964872\pi\)
							0.993917	−	0.110134i	\(-0.0351280\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.2.g.a.11.3	✓	60
3.2	odd	2	inner	69.2.g.a.11.4	yes	60
23.21	odd	22	inner	69.2.g.a.44.4	yes	60
69.44	even	22	inner	69.2.g.a.44.3	yes	60

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.2.g.a.11.3	✓	60	1.1	even	1	trivial
69.2.g.a.11.4	yes	60	3.2	odd	2	inner
69.2.g.a.44.3	yes	60	69.44	even	22	inner
69.2.g.a.44.4	yes	60	23.21	odd	22	inner