Embedded newform 980.2.x.m.863.7

• Label: 980.2.x.m.863.7
• Level: $980$
• Weight: $2$
• Character: 980.863
• Analytic conductor: $7.825$
• Analytic rank: $0$
• Dimension: $72$
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 980 = 2^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	980.x (of order \(12\), degree \(4\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.82533939809\)
Analytic rank:	\(0\)
Dimension:	\(72\)
Relative dimension:	\(18\) over \(\Q(\zeta_{12})\)
Twist minimal:	no (minimal twist has level 140)
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			863.7
Character	\(\chi\)	\(=\)	980.863
Dual form			980.2.x.m.67.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.674418 + 1.24304i) q^{2} +(-2.47915 - 0.664287i) q^{3} +(-1.09032 - 1.67666i) q^{4} +(-1.93364 - 1.12296i) q^{5} +(2.49773 - 2.63369i) q^{6} +(2.81950 - 0.224544i) q^{8} +(3.10685 + 1.79374i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q + 2 q^{2} + 8 q^{5} + 16 q^{6} - 4 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/980\mathbb{Z}\right)^\times\).

\(n\)	\(101\)	\(197\)	\(491\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{3}{4}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.674418	+	1.24304i	−0.476886	+	0.878965i
							\(3\)	−2.47915	−	0.664287i	−1.43134	−	0.383527i	−0.541848	−	0.840476i	\(-0.682275\pi\)
−0.889492	+	0.456950i	\(0.848942\pi\)
\(5\)	−1.93364	−	1.12296i	−0.864751	−	0.502202i
							\(7\)		0			0
\(11\)	−1.59653	+	0.921758i	−0.481372	+	0.277920i								−0.720988	−	0.692947i	\(-0.756312\pi\)
							0.239616	+	0.970868i	\(0.422978\pi\)
\(13\)	2.94578	+	2.94578i	0.817012	+	0.817012i	0.985674	−	0.168662i	\(-0.0539447\pi\)
							−0.168662	+	0.985674i	\(0.553945\pi\)
\(17\)	2.96933	+	0.795631i	0.720169	+	0.192969i	0.600247	−	0.799815i	\(-0.295069\pi\)
							0.119922	+	0.992783i	\(0.461736\pi\)
\(19\)	2.66374	−	4.61374i	0.611104	−	1.05846i	−0.379950	−	0.925007i	\(-0.624059\pi\)
							0.991055	−	0.133457i	\(-0.0426077\pi\)
\(23\)	0.654945	+	2.44429i	0.136565	+	0.509669i	0.999987	+	0.00518527i	\(0.00165053\pi\)
							−0.863421	+	0.504484i	\(0.831683\pi\)
\(29\)		−	6.35796i		−	1.18064i	−0.807168	−	0.590322i	\(-0.799001\pi\)
							0.807168	−	0.590322i	\(-0.200999\pi\)
\(31\)	−3.78330	+	2.18429i	−0.679500	+	0.392310i	−0.799667	−	0.600444i	\(-0.794991\pi\)
							0.120166	+	0.992754i	\(0.461657\pi\)
\(37\)	−1.03655	−	3.86848i	−0.170409	−	0.635973i	−0.997288	−	0.0735946i	\(-0.976553\pi\)
							0.826880	−	0.562379i	\(-0.190114\pi\)
\(41\)	−10.6462			−1.66266			−0.831331	−	0.555777i	\(-0.812421\pi\)
							−0.831331	+	0.555777i	\(0.812421\pi\)
\(43\)	1.02386	−	1.02386i	0.156137	−	0.156137i	−0.624715	−	0.780853i	\(-0.714785\pi\)
							0.780853	+	0.624715i	\(0.214785\pi\)
\(47\)	−0.785472	+	0.210467i	−0.114573	+	0.0306997i	−0.315650	−	0.948876i	\(-0.602223\pi\)
							0.201077	+	0.979575i	\(0.435556\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	980.2.x.m.863.7		72
4.3	odd	2	inner	980.2.x.m.863.16		72
5.2	odd	4	inner	980.2.x.m.667.2		72
7.2	even	3		980.2.k.j.883.18		36
7.3	odd	6		140.2.w.b.123.5	yes	72
7.4	even	3	inner	980.2.x.m.263.5		72
7.5	odd	6		980.2.k.k.883.18		36
7.6	odd	2		140.2.w.b.23.7	✓	72
20.7	even	4	inner	980.2.x.m.667.5		72
28.3	even	6		140.2.w.b.123.2	yes	72
28.11	odd	6	inner	980.2.x.m.263.2		72
28.19	even	6		980.2.k.k.883.10		36
28.23	odd	6		980.2.k.j.883.10		36
28.27	even	2		140.2.w.b.23.16	yes	72
35.2	odd	12		980.2.k.j.687.10		36
35.3	even	12		700.2.be.e.207.3		72
35.12	even	12		980.2.k.k.687.10		36
35.13	even	4		700.2.be.e.107.17		72
35.17	even	12		140.2.w.b.67.16	yes	72
35.24	odd	6		700.2.be.e.543.14		72
35.27	even	4		140.2.w.b.107.2	yes	72
35.32	odd	12	inner	980.2.x.m.67.16		72
35.34	odd	2		700.2.be.e.443.12		72
140.3	odd	12		700.2.be.e.207.12		72
140.27	odd	4		140.2.w.b.107.5	yes	72
140.47	odd	12		980.2.k.k.687.18		36
140.59	even	6		700.2.be.e.543.17		72
140.67	even	12	inner	980.2.x.m.67.7		72
140.83	odd	4		700.2.be.e.107.14		72
140.87	odd	12		140.2.w.b.67.7	yes	72
140.107	even	12		980.2.k.j.687.18		36
140.139	even	2		700.2.be.e.443.3		72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
140.2.w.b.23.7	✓	72	7.6	odd	2
140.2.w.b.23.16	yes	72	28.27	even	2
140.2.w.b.67.7	yes	72	140.87	odd	12
140.2.w.b.67.16	yes	72	35.17	even	12
140.2.w.b.107.2	yes	72	35.27	even	4
140.2.w.b.107.5	yes	72	140.27	odd	4
140.2.w.b.123.2	yes	72	28.3	even	6
140.2.w.b.123.5	yes	72	7.3	odd	6
700.2.be.e.107.14		72	140.83	odd	4
700.2.be.e.107.17		72	35.13	even	4
700.2.be.e.207.3		72	35.3	even	12
700.2.be.e.207.12		72	140.3	odd	12
700.2.be.e.443.3		72	140.139	even	2
700.2.be.e.443.12		72	35.34	odd	2
700.2.be.e.543.14		72	35.24	odd	6
700.2.be.e.543.17		72	140.59	even	6
980.2.k.j.687.10		36	35.2	odd	12
980.2.k.j.687.18		36	140.107	even	12
980.2.k.j.883.10		36	28.23	odd	6
980.2.k.j.883.18		36	7.2	even	3
980.2.k.k.687.10		36	35.12	even	12
980.2.k.k.687.18		36	140.47	odd	12
980.2.k.k.883.10		36	28.19	even	6
980.2.k.k.883.18		36	7.5	odd	6
980.2.x.m.67.7		72	140.67	even	12	inner
980.2.x.m.67.16		72	35.32	odd	12	inner
980.2.x.m.263.2		72	28.11	odd	6	inner
980.2.x.m.263.5		72	7.4	even	3	inner
980.2.x.m.667.2		72	5.2	odd	4	inner
980.2.x.m.667.5		72	20.7	even	4	inner
980.2.x.m.863.7		72	1.1	even	1	trivial
980.2.x.m.863.16		72	4.3	odd	2	inner