Embedded newform 983.2.c.a.2.17

• Label: 983.2.c.a.2.17
• 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			2.17
Character	\(\chi\)	\(=\)	983.2
Dual form			983.2.c.a.492.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.585179 + 1.74757i) q^{2} +(-2.62769 - 1.47627i) q^{3} +(-1.11487 - 0.840921i) q^{4} +(-1.01017 + 1.52367i) q^{5} +(4.11756 - 3.72820i) q^{6} +(0.654151 - 1.44349i) q^{7} +(-0.917117 + 0.629381i) q^{8} +(3.16483 + 5.19619i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 39690 q - 489 q^{2} - 487 q^{3} - 565 q^{4} - 485 q^{5} - 477 q^{6} - 489 q^{7} - 485 q^{8} - 558 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.585179	+	1.74757i	−0.413784	+	1.23572i	0.512431	+	0.858728i	\(0.328745\pi\)
							−0.926216	+	0.376994i	\(0.876958\pi\)
\(3\)	−2.62769	−	1.47627i	−1.51710	−	0.852326i	−0.517102	−	0.855924i	\(-0.672989\pi\)
							−0.999994	+	0.00359742i	\(0.998855\pi\)
\(5\)	−1.01017	+	1.52367i	−0.451763	+	0.681407i	−0.986528	−	0.163594i	\(-0.947691\pi\)
							0.534765	+	0.845001i	\(0.320400\pi\)
\(7\)	0.654151	−	1.44349i	0.247246	−	0.545586i	−0.745403	−	0.666614i	\(-0.767743\pi\)
							0.992649	+	0.121027i	\(0.0386189\pi\)
\(11\)	−1.88925	−	0.121046i	−0.569631	−	0.0364968i	−0.223535	−	0.974696i	\(-0.571760\pi\)
							−0.346096	+	0.938199i	\(0.612493\pi\)
\(13\)	−0.652387	−	0.753780i	−0.180940	−	0.209061i	0.658400	−	0.752668i	\(-0.271233\pi\)
							−0.839340	+	0.543607i	\(0.817058\pi\)
\(17\)	3.34411	−	1.47694i	0.811065	−	0.358211i	0.0433958	−	0.999058i	\(-0.486182\pi\)
							0.767669	+	0.640847i	\(0.221417\pi\)
\(19\)	−3.70713	−	2.83360i	−0.850474	−	0.650073i	0.0877348	−	0.996144i	\(-0.472037\pi\)
							−0.938209	+	0.346070i	\(0.887516\pi\)
\(23\)	−0.691509	−	6.52571i	−0.144190	−	1.36070i	−0.797365	−	0.603498i	\(-0.793773\pi\)
							0.653175	−	0.757207i	\(-0.273437\pi\)
\(29\)	−5.40080	+	7.81618i	−1.00290	+	1.45143i	−0.113919	+	0.993490i	\(0.536340\pi\)
							−0.888984	+	0.457938i	\(0.848588\pi\)
\(31\)	−4.93602	−	7.34270i	−0.886536	−	1.31879i	−0.947204	−	0.320630i	\(-0.896105\pi\)
							0.0606686	−	0.998158i	\(-0.480677\pi\)
\(37\)	2.92290	−	0.725010i	0.480522	−	0.119191i	0.00700484	−	0.999975i	\(-0.497770\pi\)
							0.473517	+	0.880784i	\(0.342984\pi\)
\(41\)	9.12709	+	0.819771i	1.42541	+	0.128027i	0.775195	−	0.631722i	\(-0.217652\pi\)
							0.650217	+	0.759749i	\(0.274678\pi\)
\(43\)	4.89401	+	6.70890i	0.746329	+	1.02310i	0.998464	+	0.0554058i	\(0.0176453\pi\)
							−0.252135	+	0.967692i	\(0.581133\pi\)
\(47\)	3.40076	+	11.3496i	0.496052	+	1.65550i	0.730148	+	0.683289i	\(0.239451\pi\)
							−0.234096	+	0.972214i	\(0.575213\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	983.2.c.a.2.17	✓	39690
983.492	even	491	inner	983.2.c.a.492.17	yes	39690

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
983.2.c.a.2.17	✓	39690	1.1	even	1	trivial
983.2.c.a.492.17	yes	39690	983.492	even	491	inner