Embedded newform 69.8.e.b.58.9

• Label: 69.8.e.b.58.9
• 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			58.9
Character	\(\chi\)	\(=\)	69.58
Dual form			69.8.e.b.25.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(7.48700 - 4.81160i) q^{2} +(3.84250 - 26.7252i) q^{3} +(-20.2695 + 44.3840i) q^{4} +(-178.268 + 52.3443i) q^{5} +(-99.8221 - 218.580i) q^{6} +(-622.051 - 717.885i) q^{7} +(223.922 + 1557.41i) q^{8} +(-699.470 - 205.383i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 140 q - 12 q^{2} + 378 q^{3} - 740 q^{4} + 2 q^{5} - 1755 q^{6} - 394 q^{7} - 1612 q^{8} - 10206 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	7.48700	−	4.81160i	0.661763	−	0.425289i	−0.166185	−	0.986095i	\(-0.553145\pi\)
							0.827948	+	0.560805i	\(0.189508\pi\)
\(3\)	3.84250	−	26.7252i	0.0821655	−	0.571474i
							\(5\)	−178.268	+	52.3443i	−0.637793	+	0.187273i	−0.584613	−	0.811312i	\(-0.698754\pi\)
−0.0531795	+	0.998585i	\(0.516936\pi\)
\(7\)	−622.051	−	717.885i	−0.685461	−	0.791065i	0.301250	−	0.953545i	\(-0.402596\pi\)
							−0.986712	+	0.162480i	\(0.948051\pi\)
\(11\)	6933.87	+	4456.12i	1.57073	+	1.00945i	0.979135	+	0.203209i	\(0.0651370\pi\)
							0.591593	+	0.806237i	\(0.298499\pi\)
\(13\)	6047.24	−	6978.88i	0.763405	−	0.881017i	−0.232390	−	0.972623i	\(-0.574655\pi\)
							0.995795	+	0.0916060i	\(0.0292000\pi\)
\(17\)	14892.4	+	32609.8i	0.735180	+	1.60982i	0.791323	+	0.611398i	\(0.209392\pi\)
							−0.0561430	+	0.998423i	\(0.517880\pi\)
\(19\)	−14685.8	+	32157.3i	−0.491201	+	1.07558i	0.488029	+	0.872827i	\(0.337716\pi\)
							−0.979230	+	0.202752i	\(0.935011\pi\)
\(23\)	56524.0	−	14486.5i	0.968692	−	0.248264i
							\(29\)	1074.49	+	2352.80i	0.00818105	+	0.0179140i	0.913678	−	0.406438i	\(-0.133229\pi\)
−0.905497	+	0.424352i	\(0.860502\pi\)
\(31\)	2631.36	+	18301.5i	0.0158640	+	0.110337i	0.996215	−	0.0869252i	\(-0.0277041\pi\)
							−0.980351	+	0.197262i	\(0.936795\pi\)
\(37\)	299867.	+	88049.0i	0.973247	+	0.285771i	0.729434	−	0.684051i	\(-0.239783\pi\)
							0.243813	+	0.969822i	\(0.421602\pi\)
\(41\)	−525665.	+	154349.i	−1.19115	+	0.349752i	−0.816462	−	0.577400i	\(-0.804067\pi\)
							−0.374685	+	0.927152i	\(0.622249\pi\)
\(43\)	−45405.8	+	315804.i	−0.0870907	+	0.605729i	0.898802	+	0.438354i	\(0.144438\pi\)
							−0.985893	+	0.167376i	\(0.946471\pi\)
\(47\)	1.24552e6			1.74988			0.874939	−	0.484234i	\(-0.160902\pi\)
							0.874939	+	0.484234i	\(0.160902\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.8.e.b.58.9	yes	140
23.2	even	11	inner	69.8.e.b.25.9	✓	140

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.8.e.b.25.9	✓	140	23.2	even	11	inner
69.8.e.b.58.9	yes	140	1.1	even	1	trivial