Embedded newform 69.2.g.a.44.4

• Label: 69.2.g.a.44.4
• Level: $69$
• Weight: $2$
• Character: 69.44
• 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			44.4
Character	\(\chi\)	\(=\)	69.44
Dual form			69.2.g.a.11.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0493131 - 0.0767326i) q^{2} +(-1.63918 + 0.559548i) q^{3} +(0.827374 + 1.81170i) q^{4} +(2.86859 + 0.842294i) q^{5} +(-0.0378973 + 0.153371i) q^{6} +(-1.75762 - 1.52299i) q^{7} +(0.360384 + 0.0518154i) q^{8} +(2.37381 - 1.83440i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60 q - 11 q^{3} - 10 q^{4} - 14 q^{6} - 22 q^{7} - 11 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(28\)	\(47\)
\(\chi(n)\)	\(e\left(\frac{13}{22}\right)\)	\(-1\)

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	\(-0.807915\pi\)
							0.858251	+	0.513231i	\(0.171551\pi\)
\(3\)	−1.63918	+	0.559548i	−0.946380	+	0.323055i
							\(5\)	2.86859	+	0.842294i	1.28287	+	0.376685i	0.850960	−	0.525230i	\(-0.176021\pi\)
0.431913	+	0.901915i	\(0.357839\pi\)
\(7\)	−1.75762	−	1.52299i	−0.664318	−	0.575635i	0.256062	−	0.966660i	\(-0.417575\pi\)
							−0.920380	+	0.391025i	\(0.872120\pi\)
\(11\)	−1.83549	+	1.17959i	−0.553420	+	0.355661i	−0.787267	−	0.616612i	\(-0.788505\pi\)
							0.233847	+	0.972273i	\(0.424868\pi\)
\(13\)	−2.74970	−	3.17332i	−0.762629	−	0.880120i	0.233100	−	0.972453i	\(-0.425113\pi\)
							−0.995728	+	0.0923325i	\(0.970568\pi\)
\(17\)	1.67699	−	3.67210i	0.406731	−	0.890616i	−0.589813	−	0.807540i	\(-0.700798\pi\)
							0.996543	−	0.0830757i	\(-0.0264743\pi\)
\(19\)	2.40861	−	1.09998i	0.552574	−	0.252352i	−0.119502	−	0.992834i	\(-0.538130\pi\)
							0.672076	+	0.740482i	\(0.265403\pi\)
\(23\)	3.72801	−	3.01694i	0.777344	−	0.629076i
							\(29\)	−5.44962	−	2.48876i	−1.01197	−	0.462151i	−0.160769	−	0.986992i	\(-0.551397\pi\)
−0.851200	+	0.524841i	\(0.824125\pi\)
\(31\)	−0.113954	+	0.792566i	−0.0204667	+	0.142349i	−0.997493	−	0.0707719i	\(-0.977454\pi\)
							0.977026	+	0.213121i	\(0.0683629\pi\)
\(37\)	2.32136	+	7.90582i	0.381629	+	1.29971i	0.896729	+	0.442579i	\(0.145936\pi\)
							−0.515101	+	0.857130i	\(0.672245\pi\)
\(41\)	−2.47018	+	8.41266i	−0.385777	+	1.31384i	0.506458	+	0.862265i	\(0.330955\pi\)
							−0.892235	+	0.451572i	\(0.850864\pi\)
\(43\)	−2.82529	+	0.406215i	−0.430853	+	0.0619473i	−0.354330	−	0.935121i	\(-0.615291\pi\)
							−0.0765230	+	0.997068i	\(0.524382\pi\)
\(47\)		−	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.44.4	yes	60
3.2	odd	2	inner	69.2.g.a.44.3	yes	60
23.11	odd	22	inner	69.2.g.a.11.3	✓	60
69.11	even	22	inner	69.2.g.a.11.4	yes	60

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.2.g.a.11.3	✓	60	23.11	odd	22	inner
69.2.g.a.11.4	yes	60	69.11	even	22	inner
69.2.g.a.44.3	yes	60	3.2	odd	2	inner
69.2.g.a.44.4	yes	60	1.1	even	1	trivial