Embedded newform 983.2.c.a.2.9

• Label: 983.2.c.a.2.9
• Level: $983$
• Weight: $2$
• Character: 983.2
• Analytic conductor: $7.849$
• Analytic rank: $0$
• Dimension: $39690$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 983 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	983.c (of order \(491\), degree \(490\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.84929451869\)
Analytic rank:	\(0\)
Dimension:	\(39690\)
Relative dimension:	\(81\) over \(\Q(\zeta_{491})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{491}]$

Embedding invariants

Embedding label			2.9
Character	\(\chi\)	\(=\)	983.2
Dual form			983.2.c.a.492.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.752267 + 2.24656i) q^{2} +(-0.810068 - 0.455108i) q^{3} +(-2.88443 - 2.17566i) q^{4} +(2.05128 - 3.09401i) q^{5} +(1.63182 - 1.47751i) q^{6} +(0.245279 - 0.541247i) q^{7} +(3.15078 - 2.16226i) q^{8} +(-1.11145 - 1.82484i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 39690 q - 489 q^{2} - 487 q^{3} - 565 q^{4} - 485 q^{5} - 477 q^{6} - 489 q^{7} - 485 q^{8} - 558 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/983\mathbb{Z}\right)^\times\).

\(n\)	\(5\)
\(\chi(n)\)	\(e\left(\frac{337}{491}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.752267	+	2.24656i	−0.531933	+	1.58856i	0.252008	+	0.967725i	\(0.418909\pi\)
							−0.783941	+	0.620835i	\(0.786794\pi\)
\(3\)	−0.810068	−	0.455108i	−0.467693	−	0.262757i	0.238012	−	0.971262i	\(-0.423504\pi\)
							−0.705705	+	0.708506i	\(0.749370\pi\)
\(5\)	2.05128	−	3.09401i	0.917360	−	1.38368i	−0.00609754	−	0.999981i	\(-0.501941\pi\)
							0.923458	−	0.383700i	\(-0.125350\pi\)
\(7\)	0.245279	−	0.541247i	0.0927069	−	0.204572i	−0.858721	−	0.512443i	\(-0.828741\pi\)
							0.951428	+	0.307871i	\(0.0996164\pi\)
\(11\)	4.98914	+	0.319659i	1.50428	+	0.0963809i	0.794165	−	0.607702i	\(-0.207909\pi\)
							0.710118	+	0.704083i	\(0.248642\pi\)
\(13\)	−0.149493	−	0.172727i	−0.0414618	−	0.0479058i	0.735020	−	0.678045i	\(-0.237173\pi\)
							−0.776482	+	0.630140i	\(0.782998\pi\)
\(17\)	−3.36321	+	1.48538i	−0.815698	+	0.360258i	−0.769478	−	0.638674i	\(-0.779483\pi\)
							−0.0462203	+	0.998931i	\(0.514718\pi\)
\(19\)	−5.44151	−	4.15931i	−1.24837	−	0.954211i	−0.248472	−	0.968639i	\(-0.579929\pi\)
							−0.999896	+	0.0144283i	\(0.995407\pi\)
\(23\)	−0.224605	−	2.11958i	−0.0468333	−	0.441962i	−0.992985	−	0.118239i	\(-0.962275\pi\)
							0.946152	−	0.323723i	\(-0.104935\pi\)
\(29\)	−5.50875	+	7.97241i	−1.02295	+	1.48044i	−0.152424	+	0.988315i	\(0.548708\pi\)
							−0.870526	+	0.492123i	\(0.836221\pi\)
\(31\)	−0.0372810	−	0.0554582i	−0.00669586	−	0.00996059i	0.826548	−	0.562866i	\(-0.190301\pi\)
							−0.833244	+	0.552905i	\(0.813519\pi\)
\(37\)	4.65546	−	1.15476i	0.765354	−	0.189842i	0.161430	−	0.986884i	\(-0.448389\pi\)
							0.603923	+	0.797042i	\(0.293603\pi\)
\(41\)	−7.73094	−	0.694372i	−1.20737	−	0.108443i	−0.532532	−	0.846410i	\(-0.678760\pi\)
							−0.674837	+	0.737967i	\(0.735786\pi\)
\(43\)	−7.40511	−	10.1512i	−1.12927	−	1.54805i	−0.796085	−	0.605184i	\(-0.793099\pi\)
							−0.333183	−	0.942862i	\(-0.608123\pi\)
\(47\)	−1.59031	−	5.30744i	−0.231971	−	0.774170i	−0.992215	−	0.124533i	\(-0.960257\pi\)
							0.760244	−	0.649637i	\(-0.225079\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	983.2.c.a.2.9	✓	39690
983.492	even	491	inner	983.2.c.a.492.9	yes	39690

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
983.2.c.a.2.9	✓	39690	1.1	even	1	trivial
983.2.c.a.492.9	yes	39690	983.492	even	491	inner