Embedded newform 69.8.e.b.58.8

• Label: 69.8.e.b.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.b.25.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.46982 - 0.944597i) q^{2} +(3.84250 - 26.7252i) q^{3} +(-51.9050 + 113.656i) q^{4} +(-216.769 + 63.6491i) q^{5} +(-19.5967 - 42.9109i) q^{6} +(397.194 + 458.387i) q^{7} +(62.8953 + 437.446i) q^{8} +(-699.470 - 205.383i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 140 q - 12 q^{2} + 378 q^{3} - 740 q^{4} + 2 q^{5} - 1755 q^{6} - 394 q^{7} - 1612 q^{8} - 10206 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.46982	−	0.944597i	0.129915	−	0.0834914i	−0.474069	−	0.880488i	\(-0.657215\pi\)
							0.603984	+	0.796996i	\(0.293579\pi\)
\(3\)	3.84250	−	26.7252i	0.0821655	−	0.571474i
							\(5\)	−216.769	+	63.6491i	−0.775536	+	0.227718i	−0.645468	−	0.763787i	\(-0.723338\pi\)
−0.130068	+	0.991505i	\(0.541520\pi\)
\(7\)	397.194	+	458.387i	0.437683	+	0.505113i	0.931142	−	0.364656i	\(-0.118813\pi\)
							−0.493459	+	0.869769i	\(0.664268\pi\)
\(11\)	−1244.84	−	800.013i	−0.281994	−	0.181227i	0.391996	−	0.919967i	\(-0.371785\pi\)
							−0.673990	+	0.738740i	\(0.735421\pi\)
\(13\)	5860.42	−	6763.29i	0.739822	−	0.853800i	−0.253719	−	0.967278i	\(-0.581654\pi\)
							0.993541	+	0.113478i	\(0.0361991\pi\)
\(17\)	−4630.93	−	10140.3i	−0.228611	−	0.500588i	0.760213	−	0.649673i	\(-0.225094\pi\)
							−0.988824	+	0.149085i	\(0.952367\pi\)
\(19\)	16652.2	−	36463.2i	0.556972	−	1.21960i	−0.396478	−	0.918044i	\(-0.629768\pi\)
							0.953449	−	0.301553i	\(-0.0975052\pi\)
\(23\)	−28200.9	+	51083.6i	−0.483298	+	0.875456i
							\(29\)	−25910.5	−	56736.0i	−0.197280	−	0.431982i	0.784977	−	0.619525i	\(-0.212675\pi\)
−0.982256	+	0.187543i	\(0.939948\pi\)
\(31\)	−42672.9	−	296796.i	−0.257268	−	1.78934i	−0.552087	−	0.833786i	\(-0.686168\pi\)
							0.294819	−	0.955553i	\(-0.404741\pi\)
\(37\)	−11884.2	−	3489.51i	−0.0385712	−	0.0113255i	0.262390	−	0.964962i	\(-0.415489\pi\)
							−0.300961	+	0.953636i	\(0.597307\pi\)
\(41\)	595929.	−	174980.i	1.35036	−	0.396502i	0.475008	−	0.879982i	\(-0.342445\pi\)
							0.875356	+	0.483479i	\(0.160627\pi\)
\(43\)	74147.1	−	515705.i	0.142218	−	0.989148i	−0.786296	−	0.617850i	\(-0.788004\pi\)
							0.928514	−	0.371298i	\(-0.121087\pi\)
\(47\)	116420.			0.163563			0.0817817	−	0.996650i	\(-0.473939\pi\)
							0.0817817	+	0.996650i	\(0.473939\pi\)

(See \(a_n\) instead)

Twists

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

    

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