Embedded newform 983.2.c.a.2.8

• Label: 983.2.c.a.2.8
• 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			2.8
Character	\(\chi\)	\(=\)	983.2
Dual form			983.2.c.a.492.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.752960 + 2.24863i) q^{2} +(-1.39225 - 0.782183i) q^{3} +(-2.89269 - 2.18189i) q^{4} +(0.987618 - 1.48965i) q^{5} +(2.80715 - 2.54170i) q^{6} +(-0.299880 + 0.661733i) q^{7} +(3.17392 - 2.17814i) q^{8} +(-0.233998 - 0.384191i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 39690 q - 489 q^{2} - 487 q^{3} - 565 q^{4} - 485 q^{5} - 477 q^{6} - 489 q^{7} - 485 q^{8} - 558 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.752960	+	2.24863i	−0.532423	+	1.59002i	0.250639	+	0.968081i	\(0.419359\pi\)
							−0.783062	+	0.621944i	\(0.786343\pi\)
\(3\)	−1.39225	−	0.782183i	−0.803813	−	0.451594i	0.0327504	−	0.999464i	\(-0.489573\pi\)
							−0.836564	+	0.547870i	\(0.815439\pi\)
\(5\)	0.987618	−	1.48965i	0.441676	−	0.666193i	−0.543170	−	0.839623i	\(-0.682776\pi\)
							0.984846	+	0.173429i	\(0.0554848\pi\)
\(7\)	−0.299880	+	0.661733i	−0.113344	+	0.250111i	−0.958883	−	0.283803i	\(-0.908404\pi\)
							0.845538	+	0.533915i	\(0.179280\pi\)
\(11\)	−4.39578	−	0.281642i	−1.32538	−	0.0849182i	−0.614881	−	0.788620i	\(-0.710796\pi\)
							−0.710496	+	0.703702i	\(0.751529\pi\)
\(13\)	−4.39025	−	5.07258i	−1.21764	−	1.40688i	−0.886136	−	0.463425i	\(-0.846621\pi\)
							−0.331500	−	0.943455i	\(-0.607555\pi\)
\(17\)	1.19195	−	0.526432i	0.289091	−	0.127678i	−0.254386	−	0.967103i	\(-0.581873\pi\)
							0.543477	+	0.839424i	\(0.317108\pi\)
\(19\)	3.98092	+	3.04288i	0.913286	+	0.698085i	0.953592	−	0.301102i	\(-0.0973545\pi\)
							−0.0403056	+	0.999187i	\(0.512833\pi\)
\(23\)	0.588907	+	5.55746i	0.122796	+	1.15881i	0.869596	+	0.493763i	\(0.164379\pi\)
							−0.746801	+	0.665048i	\(0.768411\pi\)
\(29\)	−3.23884	+	4.68734i	−0.601438	+	0.870417i	−0.998885	−	0.0472081i	\(-0.984968\pi\)
							0.397447	+	0.917625i	\(0.369896\pi\)
\(31\)	1.82812	+	2.71946i	0.328340	+	0.488430i	0.957335	−	0.288979i	\(-0.0933157\pi\)
							−0.628996	+	0.777409i	\(0.716534\pi\)
\(37\)	4.26298	−	1.05741i	0.700829	−	0.173837i	0.125903	−	0.992043i	\(-0.459817\pi\)
							0.574926	+	0.818206i	\(0.305031\pi\)
\(41\)	3.39662	+	0.305075i	0.530463	+	0.0476448i	0.351458	−	0.936203i	\(-0.385686\pi\)
							0.179004	+	0.983848i	\(0.442712\pi\)
\(43\)	4.06614	+	5.57403i	0.620081	+	0.850032i	0.997094	−	0.0761759i	\(-0.0242710\pi\)
							−0.377014	+	0.926208i	\(0.623049\pi\)
\(47\)	−3.46304	−	11.5574i	−0.505136	−	1.68582i	−0.707670	−	0.706543i	\(-0.750254\pi\)
							0.202534	−	0.979275i	\(-0.435082\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	983.2.c.a.2.8	✓	39690
983.492	even	491	inner	983.2.c.a.492.8	yes	39690

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
983.2.c.a.2.8	✓	39690	1.1	even	1	trivial
983.2.c.a.492.8	yes	39690	983.492	even	491	inner