Embedded newform 69.8.g.a.5.4

• Label: 69.8.g.a.5.4
• 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			5.4
Character	\(\chi\)	\(=\)	69.5
Dual form			69.8.g.a.14.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-5.50277 + 18.7407i) q^{2} +(-46.6006 - 3.92169i) q^{3} +(-213.253 - 137.049i) q^{4} +(-12.7497 + 88.6762i) q^{5} +(329.928 - 851.749i) q^{6} +(98.7958 - 45.1185i) q^{7} +(1852.45 - 1605.15i) q^{8} +(2156.24 + 365.506i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 540 q - 35 q^{3} + 3362 q^{4} - 1280 q^{6} - 22 q^{7} - 659 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−5.50277	+	18.7407i	−0.486380	+	1.65646i	0.241229	+	0.970468i	\(0.422449\pi\)
							−0.727610	+	0.685991i	\(0.759369\pi\)
\(3\)	−46.6006	−	3.92169i	−0.996478	−	0.0838587i
							\(5\)	−12.7497	+	88.6762i	−0.0456147	+	0.317258i	0.954220	+	0.299104i	\(0.0966878\pi\)
−0.999835	+	0.0181534i	\(0.994221\pi\)
\(7\)	98.7958	−	45.1185i	0.108867	−	0.0497178i	−0.360237	−	0.932861i	\(-0.617304\pi\)
							0.469104	+	0.883143i	\(0.344577\pi\)
\(11\)	5427.47	−	1593.65i	1.22948	−	0.361009i	0.398429	−	0.917199i	\(-0.369555\pi\)
							0.831055	+	0.556190i	\(0.187737\pi\)
\(13\)	336.950	−	737.818i	0.0425367	−	0.0931424i	−0.887171	−	0.461441i	\(-0.847332\pi\)
							0.929707	+	0.368299i	\(0.120060\pi\)
\(17\)	8009.44	−	5147.36i	0.395395	−	0.254105i	−0.327802	−	0.944746i	\(-0.606308\pi\)
							0.723197	+	0.690641i	\(0.242672\pi\)
\(19\)	−22298.8	+	34697.5i	−0.745835	+	1.16054i	0.236174	+	0.971711i	\(0.424106\pi\)
							−0.982009	+	0.188832i	\(0.939530\pi\)
\(23\)	27152.5	−	51648.5i	0.465332	−	0.885136i
							\(29\)	−44046.7	−	68538.0i	−0.335367	−	0.521841i	0.632084	−	0.774900i	\(-0.282200\pi\)
−0.967451	+	0.253059i	\(0.918563\pi\)
\(31\)	−61086.9	−	70498.0i	−0.368283	−	0.425022i	0.541114	−	0.840949i	\(-0.318003\pi\)
							−0.909398	+	0.415927i	\(0.863457\pi\)
\(37\)	60409.8	−	8685.62i	0.196066	−	0.0281900i	−0.0435823	−	0.999050i	\(-0.513877\pi\)
							0.239648	+	0.970860i	\(0.422968\pi\)
\(41\)	113533.	+	16323.5i	0.257263	+	0.0369889i	0.269740	−	0.962933i	\(-0.413062\pi\)
							−0.0124764	+	0.999922i	\(0.503971\pi\)
\(43\)	51079.4	+	44260.5i	0.0979729	+	0.0848940i	0.702470	−	0.711713i	\(-0.252080\pi\)
							−0.604497	+	0.796607i	\(0.706626\pi\)
\(47\)			741549.i			1.04183i	0.853608	+	0.520915i	\(0.174409\pi\)
							−0.853608	+	0.520915i	\(0.825591\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.8.g.a.5.4	✓	540
3.2	odd	2	inner	69.8.g.a.5.51	yes	540
23.14	odd	22	inner	69.8.g.a.14.51	yes	540
69.14	even	22	inner	69.8.g.a.14.4	yes	540

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.8.g.a.5.4	✓	540	1.1	even	1	trivial
69.8.g.a.5.51	yes	540	3.2	odd	2	inner
69.8.g.a.14.4	yes	540	69.14	even	22	inner
69.8.g.a.14.51	yes	540	23.14	odd	22	inner