Embedded newform 980.2.x.m.667.2

• Label: 980.2.x.m.667.2
• Level: $980$
• Weight: $2$
• Character: 980.667
• Analytic conductor: $7.825$
• Analytic rank: $0$
• Dimension: $72$
• CM: no
• 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			667.2
Character	\(\chi\)	\(=\)	980.667
Dual form			980.2.x.m.263.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.24304 - 0.674418i) q^{2} +(-0.664287 + 2.47915i) q^{3} +(1.09032 + 1.67666i) q^{4} +(1.93933 + 1.11310i) q^{5} +(2.49773 - 2.63369i) q^{6} +(-0.224544 - 2.81950i) q^{8} +(-3.10685 - 1.79374i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q + 2 q^{2} + 8 q^{5} + 16 q^{6} - 4 q^{8}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.24304	−	0.674418i	−0.878965	−	0.476886i
							\(3\)	−0.664287	+	2.47915i	−0.383527	+	1.43134i	0.456950	+	0.889492i	\(0.348942\pi\)
−0.840476	+	0.541848i	\(0.817725\pi\)
\(5\)	1.93933	+	1.11310i	0.867295	+	0.497795i
							\(7\)		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	−0.168662	−	0.985674i	\(-0.553945\pi\)
							0.985674	+	0.168662i	\(0.0539447\pi\)
\(17\)	−0.795631	+	2.96933i	−0.192969	+	0.720169i	0.799815	+	0.600247i	\(0.204931\pi\)
							−0.992783	+	0.119922i	\(0.961736\pi\)
\(19\)	−2.66374	+	4.61374i	−0.611104	+	1.05846i	0.379950	+	0.925007i	\(0.375941\pi\)
							−0.991055	+	0.133457i	\(0.957392\pi\)
\(23\)	2.44429	−	0.654945i	0.509669	−	0.136565i	0.00518527	−	0.999987i	\(-0.498349\pi\)
							0.504484	+	0.863421i	\(0.331683\pi\)
\(29\)			6.35796i			1.18064i	0.807168	+	0.590322i	\(0.200999\pi\)
							−0.807168	+	0.590322i	\(0.799001\pi\)
\(31\)	−3.78330	+	2.18429i	−0.679500	+	0.392310i	−0.799667	−	0.600444i	\(-0.794991\pi\)
							0.120166	+	0.992754i	\(0.461657\pi\)
\(37\)	3.86848	−	1.03655i	0.635973	−	0.170409i	0.0735946	−	0.997288i	\(-0.476553\pi\)
							0.562379	+	0.826880i	\(0.309886\pi\)
\(41\)	−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	\(-0.285215\pi\)
							−0.780853	+	0.624715i	\(0.785215\pi\)
\(47\)	−0.210467	−	0.785472i	−0.0306997	−	0.114573i	0.948876	−	0.315650i	\(-0.102223\pi\)
							−0.979575	+	0.201077i	\(0.935556\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
140.2.w.b.23.7	✓	72	35.13	even	4
140.2.w.b.23.16	yes	72	140.83	odd	4
140.2.w.b.67.7	yes	72	28.3	even	6
140.2.w.b.67.16	yes	72	7.3	odd	6
140.2.w.b.107.2	yes	72	7.6	odd	2
140.2.w.b.107.5	yes	72	28.27	even	2
140.2.w.b.123.2	yes	72	140.3	odd	12
140.2.w.b.123.5	yes	72	35.3	even	12
700.2.be.e.107.14		72	140.139	even	2
700.2.be.e.107.17		72	35.34	odd	2
700.2.be.e.207.3		72	35.24	odd	6
700.2.be.e.207.12		72	140.59	even	6
700.2.be.e.443.3		72	140.27	odd	4
700.2.be.e.443.12		72	35.27	even	4
700.2.be.e.543.14		72	35.17	even	12
700.2.be.e.543.17		72	140.87	odd	12
980.2.k.j.687.10		36	7.2	even	3
980.2.k.j.687.18		36	28.23	odd	6
980.2.k.j.883.10		36	140.23	even	12
980.2.k.j.883.18		36	35.23	odd	12
980.2.k.k.687.10		36	7.5	odd	6
980.2.k.k.687.18		36	28.19	even	6
980.2.k.k.883.10		36	140.103	odd	12
980.2.k.k.883.18		36	35.33	even	12
980.2.x.m.67.7		72	28.11	odd	6	inner
980.2.x.m.67.16		72	7.4	even	3	inner
980.2.x.m.263.2		72	140.123	even	12	inner
980.2.x.m.263.5		72	35.18	odd	12	inner
980.2.x.m.667.2		72	1.1	even	1	trivial
980.2.x.m.667.5		72	4.3	odd	2	inner
980.2.x.m.863.7		72	5.3	odd	4	inner
980.2.x.m.863.16		72	20.3	even	4	inner