Embedded newform 69.8.g.a.5.3

• Label: 69.8.g.a.5.3
• 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			5.3
Character	\(\chi\)	\(=\)	69.5
Dual form			69.8.g.a.14.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-5.98416 + 20.3802i) q^{2} +(-23.3594 + 40.5134i) q^{3} +(-271.861 - 174.714i) q^{4} +(41.5571 - 289.036i) q^{5} +(-685.885 - 718.507i) q^{6} +(104.054 - 47.5200i) q^{7} +(3132.84 - 2714.62i) q^{8} +(-1095.68 - 1892.74i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 540 q - 35 q^{3} + 3362 q^{4} - 1280 q^{6} - 22 q^{7} - 659 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−5.98416	+	20.3802i	−0.528930	+	1.80137i	0.0662825	+	0.997801i	\(0.478886\pi\)
							−0.595212	+	0.803569i	\(0.702932\pi\)
\(3\)	−23.3594	+	40.5134i	−0.499502	+	0.866313i
							\(5\)	41.5571	−	289.036i	0.148679	−	1.03409i	−0.769706	−	0.638399i	\(-0.779597\pi\)
0.918385	−	0.395688i	\(-0.129494\pi\)
\(7\)	104.054	−	47.5200i	0.114661	−	0.0523640i	−0.357259	−	0.934006i	\(-0.616288\pi\)
							0.471920	+	0.881642i	\(0.343561\pi\)
\(11\)	−2851.97	+	837.414i	−0.646057	+	0.189699i	−0.588315	−	0.808632i	\(-0.700208\pi\)
							−0.0577422	+	0.998332i	\(0.518390\pi\)
\(13\)	−3480.42	+	7621.06i	−0.439370	+	0.962085i	0.552344	+	0.833616i	\(0.313733\pi\)
							−0.991714	+	0.128469i	\(0.958994\pi\)
\(17\)	−10653.2	+	6846.40i	−0.525907	+	0.337980i	−0.776505	−	0.630111i	\(-0.783009\pi\)
							0.250598	+	0.968091i	\(0.419373\pi\)
\(19\)	25195.8	−	39205.5i	0.842735	−	1.31132i	−0.105716	−	0.994396i	\(-0.533713\pi\)
							0.948451	−	0.316925i	\(-0.102650\pi\)
\(23\)	49905.7	+	30236.6i	0.855268	+	0.518185i
							\(29\)	−91674.0	−	142647.i	−0.697996	−	1.08610i	−0.991495	−	0.130144i	\(-0.958456\pi\)
0.293499	−	0.955959i	\(-0.405180\pi\)
\(31\)	25176.2	+	29054.9i	0.151784	+	0.175168i	0.826549	−	0.562864i	\(-0.190301\pi\)
							−0.674766	+	0.738032i	\(0.735755\pi\)
\(37\)	327870.	−	47140.6i	1.06413	−	0.152999i	0.412060	−	0.911157i	\(-0.364809\pi\)
							0.652071	+	0.758158i	\(0.273900\pi\)
\(41\)	−361781.	−	52016.3i	−0.819789	−	0.117868i	−0.280356	−	0.959896i	\(-0.590452\pi\)
							−0.539434	+	0.842028i	\(0.681362\pi\)
\(43\)	487993.	+	422849.i	0.935997	+	0.811046i	0.982191	−	0.187887i	\(-0.0601638\pi\)
							−0.0461940	+	0.998932i	\(0.514709\pi\)
\(47\)			848920.i			1.19268i	0.802731	+	0.596341i	\(0.203379\pi\)
							−0.802731	+	0.596341i	\(0.796621\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.8.g.a.5.3	✓	540
3.2	odd	2	inner	69.8.g.a.5.52	yes	540
23.14	odd	22	inner	69.8.g.a.14.52	yes	540
69.14	even	22	inner	69.8.g.a.14.3	yes	540

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.8.g.a.5.3	✓	540	1.1	even	1	trivial
69.8.g.a.5.52	yes	540	3.2	odd	2	inner
69.8.g.a.14.3	yes	540	69.14	even	22	inner
69.8.g.a.14.52	yes	540	23.14	odd	22	inner