Embedded newform 69.8.e.a.58.8

• Label: 69.8.e.a.58.8
• 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			58.8
Character	\(\chi\)	\(=\)	69.58
Dual form			69.8.e.a.25.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.14494 - 1.37847i) q^{2} +(-3.84250 + 26.7252i) q^{3} +(-50.4725 + 110.519i) q^{4} +(-464.485 + 136.385i) q^{5} +(28.5978 + 62.6205i) q^{6} +(1029.85 + 1188.50i) q^{7} +(90.5329 + 629.670i) q^{8} +(-699.470 - 205.383i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 140 q - 16 q^{2} - 378 q^{3} - 740 q^{4} - 6 q^{5} + 1647 q^{6} + 978 q^{7} - 824 q^{8} - 10206 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.14494	−	1.37847i	0.189587	−	0.121840i	−0.442404	−	0.896816i	\(-0.645874\pi\)
							0.631991	+	0.774976i	\(0.282238\pi\)
\(3\)	−3.84250	+	26.7252i	−0.0821655	+	0.571474i
							\(5\)	−464.485	+	136.385i	−1.66179	+	0.487946i	−0.971787	−	0.235859i	\(-0.924210\pi\)
−0.690005	+	0.723805i	\(0.742392\pi\)
\(7\)	1029.85	+	1188.50i	1.13482	+	1.30966i	0.944714	+	0.327895i	\(0.106339\pi\)
							0.190111	+	0.981763i	\(0.439115\pi\)
\(11\)	3011.62	+	1935.45i	0.682223	+	0.438438i	0.835313	−	0.549774i	\(-0.185286\pi\)
							−0.153090	+	0.988212i	\(0.548923\pi\)
\(13\)	−3522.89	+	4065.63i	−0.444731	+	0.513247i	−0.933211	−	0.359328i	\(-0.883006\pi\)
							0.488481	+	0.872575i	\(0.337551\pi\)
\(17\)	−14498.6	−	31747.5i	−0.715739	−	1.56725i	−0.819781	−	0.572677i	\(-0.805905\pi\)
							0.104042	−	0.994573i	\(-0.466822\pi\)
\(19\)	−20754.7	+	45446.5i	−0.694191	+	1.52007i	0.152683	+	0.988275i	\(0.451209\pi\)
							−0.846875	+	0.531792i	\(0.821519\pi\)
\(23\)	50820.8	−	28671.8i	0.870952	−	0.491368i
							\(29\)	68687.9	+	150406.i	0.522983	+	1.14517i	0.968298	+	0.249799i	\(0.0803644\pi\)
−0.445315	+	0.895374i	\(0.646908\pi\)
\(31\)	−18300.9	−	127285.i	−0.110333	−	0.767384i	−0.967596	−	0.252505i	\(-0.918746\pi\)
							0.857262	−	0.514880i	\(-0.172163\pi\)
\(37\)	146956.	+	43150.1i	0.476958	+	0.140048i	0.511371	−	0.859360i	\(-0.329138\pi\)
							−0.0344125	+	0.999408i	\(0.510956\pi\)
\(41\)	−111352.	+	32695.9i	−0.252322	+	0.0740884i	−0.405448	−	0.914118i	\(-0.632884\pi\)
							0.153126	+	0.988207i	\(0.451066\pi\)
\(43\)	47299.1	−	328973.i	0.0907222	−	0.630987i	−0.892834	−	0.450386i	\(-0.851287\pi\)
							0.983556	−	0.180601i	\(-0.0578044\pi\)
\(47\)	677797.			0.952264			0.476132	−	0.879374i	\(-0.342038\pi\)
							0.476132	+	0.879374i	\(0.342038\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.8.e.a.58.8	yes	140
23.2	even	11	inner	69.8.e.a.25.8	✓	140

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.8.e.a.25.8	✓	140	23.2	even	11	inner
69.8.e.a.58.8	yes	140	1.1	even	1	trivial