Embedded newform 69.8.g.a.5.1

• Label: 69.8.g.a.5.1
• 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			5.1
Character	\(\chi\)	\(=\)	69.5
Dual form			69.8.g.a.14.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-6.22368 + 21.1959i) q^{2} +(46.1239 + 7.71935i) q^{3} +(-302.852 - 194.631i) q^{4} +(-24.2322 + 168.538i) q^{5} +(-450.679 + 929.594i) q^{6} +(1127.70 - 515.004i) q^{7} +(3873.27 - 3356.20i) q^{8} +(2067.82 + 712.093i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 540 q - 35 q^{3} + 3362 q^{4} - 1280 q^{6} - 22 q^{7} - 659 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−6.22368	+	21.1959i	−0.550101	+	1.87347i	−0.0673168	+	0.997732i	\(0.521444\pi\)
							−0.482784	+	0.875739i	\(0.660374\pi\)
\(3\)	46.1239	+	7.71935i	0.986283	+	0.165066i
							\(5\)	−24.2322	+	168.538i	−0.0866956	+	0.602981i	0.899441	+	0.437043i	\(0.143974\pi\)
−0.986136	+	0.165938i	\(0.946935\pi\)
\(7\)	1127.70	−	515.004i	1.24266	−	0.567502i	0.317922	−	0.948117i	\(-0.397015\pi\)
							0.924734	+	0.380615i	\(0.124288\pi\)
\(11\)	3693.05	−	1084.38i	0.836585	−	0.245644i	0.164742	−	0.986337i	\(-0.447321\pi\)
							0.671844	+	0.740693i	\(0.265503\pi\)
\(13\)	3046.57	−	6671.07i	0.384601	−	0.842158i	−0.614002	−	0.789305i	\(-0.710441\pi\)
							0.998602	−	0.0528533i	\(-0.0168316\pi\)
\(17\)	5745.62	−	3692.49i	0.283639	−	0.182284i	−0.391084	−	0.920355i	\(-0.627900\pi\)
							0.674723	+	0.738071i	\(0.264263\pi\)
\(19\)	21051.4	−	32756.6i	0.704115	−	1.09562i	−0.286385	−	0.958114i	\(-0.592454\pi\)
							0.990500	−	0.137510i	\(-0.0439099\pi\)
\(23\)	−50761.0	+	28777.5i	−0.869927	+	0.493180i
							\(29\)	27683.0	+	43075.5i	0.210775	+	0.327973i	0.930503	−	0.366283i	\(-0.119370\pi\)
−0.719728	+	0.694256i	\(0.755734\pi\)
\(31\)	−168840.	−	194852.i	−1.01791	−	1.17473i	−0.984517	−	0.175290i	\(-0.943914\pi\)
							−0.0333940	−	0.999442i	\(-0.510632\pi\)
\(37\)	−244207.	+	35111.7i	−0.792598	+	0.113958i	−0.526709	−	0.850046i	\(-0.676574\pi\)
							−0.265888	+	0.964004i	\(0.585665\pi\)
\(41\)	−58774.2	−	8450.45i	−0.133181	−	0.0191486i	0.0754018	−	0.997153i	\(-0.475976\pi\)
							−0.208583	+	0.978005i	\(0.566885\pi\)
\(43\)	366308.	+	317408.i	0.702598	+	0.608805i	0.931111	−	0.364736i	\(-0.118841\pi\)
							−0.228513	+	0.973541i	\(0.573386\pi\)
\(47\)		−	727229.i		−	1.02171i	−0.859666	−	0.510856i	\(-0.829328\pi\)
							0.859666	−	0.510856i	\(-0.170672\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.8.g.a.5.1	✓	540
3.2	odd	2	inner	69.8.g.a.5.54	yes	540
23.14	odd	22	inner	69.8.g.a.14.54	yes	540
69.14	even	22	inner	69.8.g.a.14.1	yes	540

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.8.g.a.5.1	✓	540	1.1	even	1	trivial
69.8.g.a.5.54	yes	540	3.2	odd	2	inner
69.8.g.a.14.1	yes	540	69.14	even	22	inner
69.8.g.a.14.54	yes	540	23.14	odd	22	inner