Embedded newform 89.10.f.a.11.17

• Label: 89.10.f.a.11.17
• Level: $89$
• Weight: $10$
• Character: 89.11
• Analytic conductor: $45.838$
• Analytic rank: $0$
• Dimension: $660$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 89 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	89.f (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(45.8381894186\)
Analytic rank:	\(0\)
Dimension:	\(660\)
Relative dimension:	\(66\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			11.17
Character	\(\chi\)	\(=\)	89.11
Dual form			89.10.f.a.81.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-15.6407 - 18.0504i) q^{2} +(57.7859 - 8.30835i) q^{3} +(-8.31807 + 57.8534i) q^{4} +(915.893 - 588.609i) q^{5} +(-1053.78 - 913.107i) q^{6} +(-1849.17 - 2877.36i) q^{7} +(-9113.01 + 5856.58i) q^{8} +(-15615.5 + 4585.13i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 660 q - 9 q^{2} - 825 q^{3} - 16393 q^{4} - 5127 q^{5} - 11 q^{6} - 11 q^{7} + 1015 q^{8} + 489535 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/89\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{21}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−15.6407	−	18.0504i	−0.691229	−	0.797721i	0.296310	−	0.955092i	\(-0.404244\pi\)
							−0.987540	+	0.157371i	\(0.949698\pi\)
\(3\)	57.7859	−	8.30835i	0.411885	−	0.0592201i	0.0667421	−	0.997770i	\(-0.478740\pi\)
							0.345143	+	0.938550i	\(0.387830\pi\)
\(5\)	915.893	−	588.609i	0.655360	−	0.421174i	−0.170262	−	0.985399i	\(-0.554461\pi\)
							0.825621	+	0.564225i	\(0.190825\pi\)
\(7\)	−1849.17	−	2877.36i	−0.291095	−	0.452953i	0.664648	−	0.747157i	\(-0.268582\pi\)
							−0.955742	+	0.294204i	\(0.904945\pi\)
\(11\)	43562.6	+	27996.0i	0.897113	+	0.576539i	0.905933	−	0.423422i	\(-0.139171\pi\)
							−0.00881992	+	0.999961i	\(0.502808\pi\)
\(13\)	33207.0	−	4774.44i	0.322466	−	0.0463636i	0.0208190	−	0.999783i	\(-0.493373\pi\)
							0.301647	+	0.953420i	\(0.402464\pi\)
\(17\)	−146569.	+	169150.i	−0.425621	+	0.491193i	−0.927541	−	0.373721i	\(-0.878082\pi\)
							0.501920	+	0.864914i	\(0.332627\pi\)
\(19\)	10374.0	+	35330.5i	0.0182623	+	0.0621955i	0.968120	−	0.250487i	\(-0.0805907\pi\)
							−0.949858	+	0.312682i	\(0.898772\pi\)
\(23\)	44964.0	+	153133.i	0.0335035	+	0.114102i	0.974547	−	0.224183i	\(-0.0719713\pi\)
							−0.941044	+	0.338285i	\(0.890153\pi\)
\(29\)	−235894.	−	367058.i	−0.0619336	−	0.0963705i	0.808906	−	0.587938i	\(-0.200060\pi\)
							−0.870839	+	0.491568i	\(0.836424\pi\)
\(31\)	−2.18650e6	+	7.44653e6i	−0.425228	+	1.44819i	0.416919	+	0.908944i	\(0.363110\pi\)
							−0.842147	+	0.539249i	\(0.818708\pi\)
\(37\)			1.42848e7i			1.25305i	0.779402	+	0.626524i	\(0.215523\pi\)
							−0.779402	+	0.626524i	\(0.784477\pi\)
\(41\)	−7.71428e6	−	1.10915e6i	−0.426352	−	0.0613001i	−0.0742007	−	0.997243i	\(-0.523641\pi\)
							−0.352151	+	0.935943i	\(0.614550\pi\)
\(43\)	191526.	−	298020.i	0.00854318	−	0.0132934i	−0.836956	−	0.547271i	\(-0.815667\pi\)
							0.845499	+	0.533977i	\(0.179303\pi\)
\(47\)	3.64826e6	−	2.53742e7i	0.109055	−	0.758494i	−0.859758	−	0.510701i	\(-0.829386\pi\)
							0.968813	−	0.247792i	\(-0.0797051\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	89.10.f.a.11.17	✓	660
89.81	even	22	inner	89.10.f.a.81.17	yes	660

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
89.10.f.a.11.17	✓	660	1.1	even	1	trivial
89.10.f.a.81.17	yes	660	89.81	even	22	inner