Embedded newform 983.2.c.a.2.2

• Label: 983.2.c.a.2.2
• 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			2.2
Character	\(\chi\)	\(=\)	983.2
Dual form			983.2.c.a.492.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.882011 + 2.63403i) q^{2} +(-2.15731 - 1.21201i) q^{3} +(-4.56345 - 3.44211i) q^{4} +(-0.627390 + 0.946311i) q^{5} +(5.09524 - 4.61342i) q^{6} +(1.88896 - 4.16828i) q^{7} +(8.51098 - 5.84075i) q^{8} +(1.62450 + 2.66720i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 39690 q - 489 q^{2} - 487 q^{3} - 565 q^{4} - 485 q^{5} - 477 q^{6} - 489 q^{7} - 485 q^{8} - 558 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.882011	+	2.63403i	−0.623676	+	1.86254i	−0.133190	+	0.991091i	\(0.542522\pi\)
							−0.490486	+	0.871449i	\(0.663181\pi\)
\(3\)	−2.15731	−	1.21201i	−1.24553	−	0.699754i	−0.279985	−	0.960004i	\(-0.590329\pi\)
							−0.965541	+	0.260250i	\(0.916195\pi\)
\(5\)	−0.627390	+	0.946311i	−0.280577	+	0.423203i	−0.946444	−	0.322868i	\(-0.895353\pi\)
							0.665867	+	0.746071i	\(0.268062\pi\)
\(7\)	1.88896	−	4.16828i	0.713958	−	1.57546i	−0.100299	−	0.994957i	\(-0.531980\pi\)
							0.814258	−	0.580503i	\(-0.197144\pi\)
\(11\)	3.56400	+	0.228349i	1.07459	+	0.0688498i	0.591174	−	0.806544i	\(-0.298665\pi\)
							0.483411	+	0.875394i	\(0.339398\pi\)
\(13\)	−4.54880	−	5.25576i	−1.26161	−	1.45769i	−0.832102	−	0.554623i	\(-0.812862\pi\)
							−0.429507	−	0.903064i	\(-0.641313\pi\)
\(17\)	−3.62737	+	1.60205i	−0.879766	+	0.388554i	−0.794112	−	0.607771i	\(-0.792064\pi\)
							−0.0856539	+	0.996325i	\(0.527298\pi\)
\(19\)	0.610047	+	0.466299i	0.139954	+	0.106976i	0.674897	−	0.737912i	\(-0.264188\pi\)
							−0.534942	+	0.844888i	\(0.679667\pi\)
\(23\)	−0.602297	−	5.68382i	−0.125588	−	1.18516i	−0.861383	−	0.507956i	\(-0.830401\pi\)
							0.735796	−	0.677204i	\(-0.236808\pi\)
\(29\)	5.29720	−	7.66624i	0.983665	−	1.42359i	0.0792875	−	0.996852i	\(-0.474735\pi\)
							0.904377	−	0.426734i	\(-0.140336\pi\)
\(31\)	0.235219	+	0.349906i	0.0422467	+	0.0628451i	0.850439	−	0.526073i	\(-0.176336\pi\)
							−0.808193	+	0.588918i	\(0.799554\pi\)
\(37\)	−2.15666	+	0.534947i	−0.354552	+	0.0879448i	−0.413975	−	0.910288i	\(-0.635860\pi\)
							0.0594231	+	0.998233i	\(0.481074\pi\)
\(41\)	−3.92396	−	0.352439i	−0.612819	−	0.0550418i	−0.221291	−	0.975208i	\(-0.571027\pi\)
							−0.391528	+	0.920166i	\(0.628054\pi\)
\(43\)	1.50525	+	2.06346i	0.229549	+	0.314675i	0.907193	−	0.420715i	\(-0.138221\pi\)
							−0.677644	+	0.735390i	\(0.736999\pi\)
\(47\)	0.417279	+	1.39261i	0.0608664	+	0.203133i	0.982955	−	0.183846i	\(-0.0588547\pi\)
							−0.922089	+	0.386979i	\(0.873519\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	983.2.c.a.2.2	✓	39690
983.492	even	491	inner	983.2.c.a.492.2	yes	39690

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
983.2.c.a.2.2	✓	39690	1.1	even	1	trivial
983.2.c.a.492.2	yes	39690	983.492	even	491	inner