Embedded newform 980.2.x.m.67.15

• Label: 980.2.x.m.67.15
• Level: $980$
• Weight: $2$
• Character: 980.67
• 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			67.15
Character	\(\chi\)	\(=\)	980.67
Dual form			980.2.x.m.863.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.18977 + 0.764487i) q^{2} +(2.38471 - 0.638980i) q^{3} +(0.831118 + 1.81913i) q^{4} +(0.525600 - 2.17342i) q^{5} +(3.32575 + 1.06284i) q^{6} +(-0.401862 + 2.79973i) q^{8} +(2.68045 - 1.54756i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q + 2 q^{2} + 8 q^{5} + 16 q^{6} - 4 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(197\)	\(491\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{4}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.18977	+	0.764487i	0.841296	+	0.540574i
							\(3\)	2.38471	−	0.638980i	1.37681	−	0.368915i	0.506849	−	0.862035i	\(-0.330810\pi\)
0.869962	+	0.493119i	\(0.164143\pi\)
\(5\)	0.525600	−	2.17342i	0.235055	−	0.971982i
							\(7\)		0			0
\(11\)	4.09588	+	2.36476i	1.23496	+	0.713002i								0.968059	−	0.250723i	\(-0.0806684\pi\)
							0.266897	+	0.963725i	\(0.414002\pi\)
\(13\)	−0.0592655	+	0.0592655i	−0.0164373	+	0.0164373i	−0.715278	−	0.698840i	\(-0.753700\pi\)
							0.698840	+	0.715278i	\(0.253700\pi\)
\(17\)	−4.77484	+	1.27942i	−1.15807	+	0.310304i	−0.786195	−	0.617979i	\(-0.787952\pi\)
							−0.371875	+	0.928283i	\(0.621285\pi\)
\(19\)	−1.31544	−	2.27842i	−0.301784	−	0.522705i	0.674756	−	0.738041i	\(-0.264249\pi\)
							−0.976540	+	0.215336i	\(0.930915\pi\)
\(23\)	0.292774	−	1.09265i	0.0610475	−	0.227832i	−0.928661	−	0.370929i	\(-0.879039\pi\)
							0.989709	+	0.143097i	\(0.0457060\pi\)
\(29\)		−	7.27332i		−	1.35062i	−0.737533	−	0.675311i	\(-0.764009\pi\)
							0.737533	−	0.675311i	\(-0.235991\pi\)
\(31\)	−4.01558	−	2.31840i	−0.721219	−	0.416396i	0.0939821	−	0.995574i	\(-0.470040\pi\)
							−0.815201	+	0.579178i	\(0.803374\pi\)
\(37\)	−0.596933	+	2.22779i	−0.0981352	+	0.366246i	−0.997476	−	0.0710024i	\(-0.977380\pi\)
							0.899341	+	0.437248i	\(0.144047\pi\)
\(41\)	5.71767			0.892950			0.446475	−	0.894796i	\(-0.352679\pi\)
							0.446475	+	0.894796i	\(0.352679\pi\)
\(43\)	1.57302	+	1.57302i	0.239883	+	0.239883i	0.816802	−	0.576918i	\(-0.195745\pi\)
							−0.576918	+	0.816802i	\(0.695745\pi\)
\(47\)	2.67468	+	0.716679i	0.390143	+	0.104538i	0.448558	−	0.893754i	\(-0.351938\pi\)
							−0.0584148	+	0.998292i	\(0.518605\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	980.2.x.m.67.15		72
4.3	odd	2	inner	980.2.x.m.67.1		72
5.3	odd	4	inner	980.2.x.m.263.12		72
7.2	even	3	inner	980.2.x.m.667.9		72
7.3	odd	6		980.2.k.k.687.4		36
7.4	even	3		980.2.k.j.687.4		36
7.5	odd	6		140.2.w.b.107.9	yes	72
7.6	odd	2		140.2.w.b.67.15	yes	72
20.3	even	4	inner	980.2.x.m.263.9		72
28.3	even	6		980.2.k.k.687.14		36
28.11	odd	6		980.2.k.j.687.14		36
28.19	even	6		140.2.w.b.107.12	yes	72
28.23	odd	6	inner	980.2.x.m.667.12		72
28.27	even	2		140.2.w.b.67.1	yes	72
35.3	even	12		980.2.k.k.883.14		36
35.12	even	12		700.2.be.e.443.18		72
35.13	even	4		140.2.w.b.123.12	yes	72
35.18	odd	12		980.2.k.j.883.14		36
35.19	odd	6		700.2.be.e.107.10		72
35.23	odd	12	inner	980.2.x.m.863.1		72
35.27	even	4		700.2.be.e.543.7		72
35.33	even	12		140.2.w.b.23.1	✓	72
35.34	odd	2		700.2.be.e.207.4		72
140.3	odd	12		980.2.k.k.883.4		36
140.19	even	6		700.2.be.e.107.7		72
140.23	even	12	inner	980.2.x.m.863.15		72
140.27	odd	4		700.2.be.e.543.10		72
140.47	odd	12		700.2.be.e.443.4		72
140.83	odd	4		140.2.w.b.123.9	yes	72
140.103	odd	12		140.2.w.b.23.15	yes	72
140.123	even	12		980.2.k.j.883.4		36
140.139	even	2		700.2.be.e.207.18		72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
140.2.w.b.23.1	✓	72	35.33	even	12
140.2.w.b.23.15	yes	72	140.103	odd	12
140.2.w.b.67.1	yes	72	28.27	even	2
140.2.w.b.67.15	yes	72	7.6	odd	2
140.2.w.b.107.9	yes	72	7.5	odd	6
140.2.w.b.107.12	yes	72	28.19	even	6
140.2.w.b.123.9	yes	72	140.83	odd	4
140.2.w.b.123.12	yes	72	35.13	even	4
700.2.be.e.107.7		72	140.19	even	6
700.2.be.e.107.10		72	35.19	odd	6
700.2.be.e.207.4		72	35.34	odd	2
700.2.be.e.207.18		72	140.139	even	2
700.2.be.e.443.4		72	140.47	odd	12
700.2.be.e.443.18		72	35.12	even	12
700.2.be.e.543.7		72	35.27	even	4
700.2.be.e.543.10		72	140.27	odd	4
980.2.k.j.687.4		36	7.4	even	3
980.2.k.j.687.14		36	28.11	odd	6
980.2.k.j.883.4		36	140.123	even	12
980.2.k.j.883.14		36	35.18	odd	12
980.2.k.k.687.4		36	7.3	odd	6
980.2.k.k.687.14		36	28.3	even	6
980.2.k.k.883.4		36	140.3	odd	12
980.2.k.k.883.14		36	35.3	even	12
980.2.x.m.67.1		72	4.3	odd	2	inner
980.2.x.m.67.15		72	1.1	even	1	trivial
980.2.x.m.263.9		72	20.3	even	4	inner
980.2.x.m.263.12		72	5.3	odd	4	inner
980.2.x.m.667.9		72	7.2	even	3	inner
980.2.x.m.667.12		72	28.23	odd	6	inner
980.2.x.m.863.1		72	35.23	odd	12	inner
980.2.x.m.863.15		72	140.23	even	12	inner