Embedded newform 983.2.c.a.2.3

• Label: 983.2.c.a.2.3
• 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			2.3
Character	\(\chi\)	\(=\)	983.2
Dual form			983.2.c.a.492.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.873868 + 2.60971i) q^{2} +(-2.35753 - 1.32449i) q^{3} +(-4.45024 - 3.35672i) q^{4} +(0.433776 - 0.654278i) q^{5} +(5.51671 - 4.99503i) q^{6} +(-1.45490 + 3.21047i) q^{7} +(8.11063 - 5.56601i) q^{8} +(2.24312 + 3.68287i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 39690 q - 489 q^{2} - 487 q^{3} - 565 q^{4} - 485 q^{5} - 477 q^{6} - 489 q^{7} - 485 q^{8} - 558 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.873868	+	2.60971i	−0.617918	+	1.84535i	−0.0902044	+	0.995923i	\(0.528752\pi\)
							−0.527714	+	0.849422i	\(0.676951\pi\)
\(3\)	−2.35753	−	1.32449i	−1.36112	−	0.764696i	−0.374420	−	0.927259i	\(-0.622158\pi\)
							−0.986698	+	0.162563i	\(0.948024\pi\)
\(5\)	0.433776	−	0.654278i	0.193991	−	0.292602i	−0.723528	−	0.690295i	\(-0.757481\pi\)
							0.917518	+	0.397694i	\(0.130189\pi\)
\(7\)	−1.45490	+	3.21047i	−0.549902	+	1.21344i	0.404395	+	0.914584i	\(0.367482\pi\)
							−0.954297	+	0.298860i	\(0.903394\pi\)
\(11\)	−2.67578	−	0.171440i	−0.806778	−	0.0516911i	−0.344906	−	0.938637i	\(-0.612089\pi\)
							−0.461873	+	0.886946i	\(0.652822\pi\)
\(13\)	3.71794	+	4.29577i	1.03117	+	1.19143i	0.981262	+	0.192679i	\(0.0617176\pi\)
							0.0499079	+	0.998754i	\(0.484107\pi\)
\(17\)	−2.36144	+	1.04294i	−0.572733	+	0.252951i	−0.670066	−	0.742301i	\(-0.733734\pi\)
							0.0973337	+	0.995252i	\(0.468969\pi\)
\(19\)	−5.97309	−	4.56563i	−1.37032	−	1.04743i	−0.992590	−	0.121510i	\(-0.961226\pi\)
							−0.377730	−	0.925916i	\(-0.623295\pi\)
\(23\)	0.529157	+	4.99361i	0.110337	+	1.04124i	0.902429	+	0.430839i	\(0.141782\pi\)
							−0.792092	+	0.610402i	\(0.791008\pi\)
\(29\)	−1.27951	+	1.85174i	−0.237599	+	0.343859i	−0.923338	−	0.383988i	\(-0.874550\pi\)
							0.685739	+	0.727847i	\(0.259479\pi\)
\(31\)	−4.36425	−	6.49215i	−0.783843	−	1.16602i	−0.982556	−	0.185967i	\(-0.940458\pi\)
							0.198713	−	0.980058i	\(-0.436324\pi\)
\(37\)	−0.993452	+	0.246420i	−0.163322	+	0.0405113i	−0.321557	−	0.946890i	\(-0.604206\pi\)
							0.158234	+	0.987402i	\(0.449420\pi\)
\(41\)	6.68351	+	0.600295i	1.04379	+	0.0937503i	0.598087	−	0.801431i	\(-0.295928\pi\)
							0.445702	+	0.895182i	\(0.352954\pi\)
\(43\)	3.20565	+	4.39444i	0.488857	+	0.670146i	0.979553	−	0.201185i	\(-0.0644794\pi\)
							−0.490696	+	0.871331i	\(0.663257\pi\)
\(47\)	−1.38812	−	4.63266i	−0.202479	−	0.675743i	−0.997597	−	0.0692880i	\(-0.977927\pi\)
							0.795118	−	0.606455i	\(-0.207409\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	983.2.c.a.2.3	✓	39690
983.492	even	491	inner	983.2.c.a.492.3	yes	39690

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
983.2.c.a.2.3	✓	39690	1.1	even	1	trivial
983.2.c.a.492.3	yes	39690	983.492	even	491	inner