Embedded newform 69.8.g.a.5.15

• Label: 69.8.g.a.5.15
• 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			5.15
Character	\(\chi\)	\(=\)	69.5
Dual form			69.8.g.a.14.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.58658 + 12.2148i) q^{2} +(-5.91340 + 46.3900i) q^{3} +(-28.6569 - 18.4167i) q^{4} +(47.6282 - 331.261i) q^{5} +(-545.435 - 238.613i) q^{6} +(343.311 - 156.785i) q^{7} +(-903.757 + 783.110i) q^{8} +(-2117.06 - 548.645i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 540 q - 35 q^{3} + 3362 q^{4} - 1280 q^{6} - 22 q^{7} - 659 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.58658	+	12.2148i	−0.317012	+	1.07964i	0.634727	+	0.772736i	\(0.281113\pi\)
							−0.951739	+	0.306908i	\(0.900706\pi\)
\(3\)	−5.91340	+	46.3900i	−0.126448	+	0.991973i
							\(5\)	47.6282	−	331.261i	0.170400	−	1.18516i	−0.707641	−	0.706572i	\(-0.750241\pi\)
0.878041	−	0.478585i	\(-0.158850\pi\)
\(7\)	343.311	−	156.785i	0.378307	−	0.172767i	−0.217181	−	0.976131i	\(-0.569686\pi\)
							0.595488	+	0.803364i	\(0.296959\pi\)
\(11\)	7136.26	−	2095.40i	1.61658	−	0.474670i	0.656482	−	0.754342i	\(-0.272044\pi\)
							0.960096	+	0.279672i	\(0.0902256\pi\)
\(13\)	4074.17	−	8921.20i	0.514325	−	1.12622i	−0.457218	−	0.889354i	\(-0.651154\pi\)
							0.971544	−	0.236861i	\(-0.0761185\pi\)
\(17\)	24291.6	−	15611.3i	1.19918	−	0.770667i	0.220367	−	0.975417i	\(-0.429275\pi\)
							0.978814	+	0.204750i	\(0.0656383\pi\)
\(19\)	−2636.55	+	4102.56i	−0.0881859	+	0.137220i	−0.882544	−	0.470230i	\(-0.844171\pi\)
							0.794358	+	0.607450i	\(0.207807\pi\)
\(23\)	−22469.1	+	53851.3i	−0.385069	+	0.922888i
							\(29\)	29583.0	+	46032.1i	0.225242	+	0.350484i	0.935420	−	0.353539i	\(-0.115022\pi\)
−0.710178	+	0.704022i	\(0.751385\pi\)
\(31\)	−140328.	−	161948.i	−0.846018	−	0.976357i	0.153913	−	0.988084i	\(-0.450812\pi\)
							−0.999931	+	0.0117275i	\(0.996267\pi\)
\(37\)	45755.4	−	6578.64i	0.148504	−	0.0213516i	−0.0676621	−	0.997708i	\(-0.521554\pi\)
							0.216166	+	0.976357i	\(0.430645\pi\)
\(41\)	522867.	+	75176.9i	1.18481	+	0.170350i	0.706416	−	0.707797i	\(-0.250311\pi\)
							0.478391	+	0.878147i	\(0.341220\pi\)
\(43\)	−578772.	−	501509.i	−1.11011	−	0.961920i	−0.110625	−	0.993862i	\(-0.535285\pi\)
							−0.999490	+	0.0319427i	\(0.989831\pi\)
\(47\)			66342.2i			0.0932068i	0.998913	+	0.0466034i	\(0.0148397\pi\)
							−0.998913	+	0.0466034i	\(0.985160\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.8.g.a.5.15	✓	540
3.2	odd	2	inner	69.8.g.a.5.40	yes	540
23.14	odd	22	inner	69.8.g.a.14.40	yes	540
69.14	even	22	inner	69.8.g.a.14.15	yes	540

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.8.g.a.5.15	✓	540	1.1	even	1	trivial
69.8.g.a.5.40	yes	540	3.2	odd	2	inner
69.8.g.a.14.15	yes	540	69.14	even	22	inner
69.8.g.a.14.40	yes	540	23.14	odd	22	inner