Embedded newform 69.8.e.b.58.3

• Label: 69.8.e.b.58.3
• Level: $69$
• Weight: $8$
• Character: 69.58
• Analytic conductor: $21.555$
• Analytic rank: $0$
• Dimension: $140$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 69 = 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	69.e (of order \(11\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(21.5545667584\)
Analytic rank:	\(0\)
Dimension:	\(140\)
Relative dimension:	\(14\) over \(\Q(\zeta_{11})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{11}]$

Embedding invariants

Embedding label			58.3
Character	\(\chi\)	\(=\)	69.58
Dual form			69.8.e.b.25.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-13.9010 + 8.93366i) q^{2} +(3.84250 - 26.7252i) q^{3} +(60.2557 - 131.942i) q^{4} +(-470.340 + 138.104i) q^{5} +(185.339 + 405.836i) q^{6} +(934.970 + 1079.01i) q^{7} +(40.0940 + 278.860i) q^{8} +(-699.470 - 205.383i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 140 q - 12 q^{2} + 378 q^{3} - 740 q^{4} + 2 q^{5} - 1755 q^{6} - 394 q^{7} - 1612 q^{8} - 10206 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/69\mathbb{Z}\right)^\times\).

\(n\)	\(28\)	\(47\)
\(\chi(n)\)	\(e\left(\frac{10}{11}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−13.9010	+	8.93366i	−1.22869	+	0.789631i	−0.983687	−	0.179891i	\(-0.942426\pi\)
							−0.245004	+	0.969522i	\(0.578789\pi\)
\(3\)	3.84250	−	26.7252i	0.0821655	−	0.571474i
							\(5\)	−470.340	+	138.104i	−1.68274	+	0.494097i	−0.976796	−	0.214172i	\(-0.931295\pi\)
−0.705943	+	0.708269i	\(0.749477\pi\)
\(7\)	934.970	+	1079.01i	1.03028	+	1.18900i	0.981749	+	0.190182i	\(0.0609079\pi\)
							0.0485292	+	0.998822i	\(0.484547\pi\)
\(11\)	4321.84	+	2777.48i	0.979025	+	0.629181i	0.929200	−	0.369577i	\(-0.120497\pi\)
							0.0498250	+	0.998758i	\(0.484134\pi\)
\(13\)	5642.01	−	6511.23i	0.712250	−	0.821980i	−0.278103	−	0.960551i	\(-0.589706\pi\)
							0.990352	+	0.138571i	\(0.0442511\pi\)
\(17\)	6479.70	+	14188.6i	0.319877	+	0.700434i	0.999450	−	0.0331750i	\(-0.0105619\pi\)
							−0.679572	+	0.733609i	\(0.737835\pi\)
\(19\)	−12802.2	+	28032.8i	−0.428199	+	0.937625i	0.565417	+	0.824805i	\(0.308715\pi\)
							−0.993616	+	0.112819i	\(0.964012\pi\)
\(23\)	−57004.7	+	12461.4i	−0.976930	+	0.213559i
							\(29\)	−54016.7	−	118280.i	−0.411278	−	0.900573i	−0.996001	−	0.0893403i	\(-0.971524\pi\)
0.584723	−	0.811233i	\(-0.301203\pi\)
\(31\)	27073.3	+	188299.i	0.163221	+	1.13523i	0.892512	+	0.451023i	\(0.148941\pi\)
							−0.729292	+	0.684203i	\(0.760150\pi\)
\(37\)	−243682.	−	71551.6i	−0.790893	−	0.232227i	−0.138755	−	0.990327i	\(-0.544310\pi\)
							−0.652139	+	0.758100i	\(0.726128\pi\)
\(41\)	−417035.	+	122453.i	−0.944994	+	0.277475i	−0.717701	−	0.696351i	\(-0.754806\pi\)
							−0.227293	+	0.973826i	\(0.572987\pi\)
\(43\)	67029.9	−	466204.i	0.128567	−	0.894203i	−0.818806	−	0.574070i	\(-0.805364\pi\)
							0.947373	−	0.320132i	\(-0.103727\pi\)
\(47\)	31114.9			0.0437146			0.0218573	−	0.999761i	\(-0.493042\pi\)
							0.0218573	+	0.999761i	\(0.493042\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.8.e.b.58.3	yes	140
23.2	even	11	inner	69.8.e.b.25.3	✓	140

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.8.e.b.25.3	✓	140	23.2	even	11	inner
69.8.e.b.58.3	yes	140	1.1	even	1	trivial