Embedded newform 69.8.g.a.5.5

• Label: 69.8.g.a.5.5
• Level: $69$
• Weight: $8$
• Character: 69.5
• Analytic conductor: $21.555$
• Analytic rank: $0$
• Dimension: $540$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 69 = 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	69.g (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(21.5545667584\)
Analytic rank:	\(0\)
Dimension:	\(540\)
Relative dimension:	\(54\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			5.5
Character	\(\chi\)	\(=\)	69.5
Dual form			69.8.g.a.14.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-5.36309 + 18.2650i) q^{2} +(-20.6078 - 41.9800i) q^{3} +(-197.168 - 126.712i) q^{4} +(51.4914 - 358.131i) q^{5} +(877.287 - 151.259i) q^{6} +(605.621 - 276.578i) q^{7} +(1530.35 - 1326.05i) q^{8} +(-1337.64 + 1730.23i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 540 q - 35 q^{3} + 3362 q^{4} - 1280 q^{6} - 22 q^{7} - 659 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−5.36309	+	18.2650i	−0.474035	+	1.61441i	0.281676	+	0.959509i	\(0.409110\pi\)
							−0.755711	+	0.654905i	\(0.772709\pi\)
\(3\)	−20.6078	−	41.9800i	−0.440663	−	0.897673i
							\(5\)	51.4914	−	358.131i	0.184221	−	1.28129i	−0.662424	−	0.749129i	\(-0.730472\pi\)
0.846645	−	0.532158i	\(-0.178619\pi\)
\(7\)	605.621	−	276.578i	0.667357	−	0.304772i	−0.0527775	−	0.998606i	\(-0.516807\pi\)
							0.720134	+	0.693835i	\(0.244080\pi\)
\(11\)	−1242.90	+	364.948i	−0.281554	+	0.0826717i	−0.419461	−	0.907773i	\(-0.637781\pi\)
							0.137907	+	0.990445i	\(0.455963\pi\)
\(13\)	4209.00	−	9216.43i	0.531346	−	1.16349i	−0.433617	−	0.901097i	\(-0.642763\pi\)
							0.964963	−	0.262388i	\(-0.0845099\pi\)
\(17\)	−14800.1	+	9511.46i	−0.730624	+	0.469544i	−0.852318	−	0.523024i	\(-0.824804\pi\)
							0.121694	+	0.992568i	\(0.461167\pi\)
\(19\)	4199.58	−	6534.68i	0.140465	−	0.218568i	−0.763892	−	0.645344i	\(-0.776714\pi\)
							0.904358	+	0.426776i	\(0.140351\pi\)
\(23\)	−56493.6	+	14604.7i	−0.968171	+	0.250291i
							\(29\)	−22064.0	−	34332.3i	−0.167993	−	0.261403i	0.747042	−	0.664777i	\(-0.231474\pi\)
−0.915035	+	0.403374i	\(0.867837\pi\)
\(31\)	188561.	+	217611.i	1.13681	+	1.31195i	0.943713	+	0.330765i	\(0.107307\pi\)
							0.193094	+	0.981180i	\(0.438148\pi\)
\(37\)	−412813.	+	59353.6i	−1.33982	+	0.192638i	−0.774679	−	0.632354i	\(-0.782089\pi\)
							−0.565145	+	0.824992i	\(0.691180\pi\)
\(41\)	−376178.	−	54086.2i	−0.852413	−	0.122558i	−0.297760	−	0.954641i	\(-0.596240\pi\)
							−0.554652	+	0.832082i	\(0.687149\pi\)
\(43\)	−726905.	−	629867.i	−1.39424	−	1.20812i	−0.950015	−	0.312203i	\(-0.898933\pi\)
							−0.444227	−	0.895914i	\(-0.646522\pi\)
\(47\)		−	285625.i		−	0.401286i	−0.979664	−	0.200643i	\(-0.935697\pi\)
							0.979664	−	0.200643i	\(-0.0643032\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.8.g.a.5.5	✓	540
3.2	odd	2	inner	69.8.g.a.5.50	yes	540
23.14	odd	22	inner	69.8.g.a.14.50	yes	540
69.14	even	22	inner	69.8.g.a.14.5	yes	540

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.8.g.a.5.5	✓	540	1.1	even	1	trivial
69.8.g.a.5.50	yes	540	3.2	odd	2	inner
69.8.g.a.14.5	yes	540	69.14	even	22	inner
69.8.g.a.14.50	yes	540	23.14	odd	22	inner