Embedded newform 980.2.x.m.667.18

• Label: 980.2.x.m.667.18
• 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			667.18
Character	\(\chi\)	\(=\)	980.667
Dual form			980.2.x.m.263.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.41349 - 0.0450897i) q^{2} +(0.543458 - 2.02821i) q^{3} +(1.99593 - 0.127468i) q^{4} +(1.62425 - 1.53682i) q^{5} +(0.676724 - 2.89137i) q^{6} +(2.81549 - 0.270171i) q^{8} +(-1.22023 - 0.704499i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q + 2 q^{2} + 8 q^{5} + 16 q^{6} - 4 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.41349	−	0.0450897i	0.999492	−	0.0318832i
							\(3\)	0.543458	−	2.02821i	0.313766	−	1.17099i	−0.611367	−	0.791347i	\(-0.709380\pi\)
0.925133	−	0.379643i	\(-0.123953\pi\)
\(5\)	1.62425	−	1.53682i	0.726386	−	0.687287i
							\(7\)		0			0
\(11\)	0.366133	−	0.211387i	0.110393	−	0.0637356i								−0.443787	−	0.896132i	\(-0.646365\pi\)
							0.554180	+	0.832397i	\(0.313032\pi\)
\(13\)	1.56422	−	1.56422i	0.433837	−	0.433837i	−0.456095	−	0.889931i	\(-0.650752\pi\)
							0.889931	+	0.456095i	\(0.150752\pi\)
\(17\)	−1.18219	+	4.41198i	−0.286722	+	1.07006i	0.660849	+	0.750519i	\(0.270196\pi\)
							−0.947572	+	0.319544i	\(0.896470\pi\)
\(19\)	−3.66683	+	6.35114i	−0.841228	+	1.45705i	0.0476283	+	0.998865i	\(0.484834\pi\)
							−0.888857	+	0.458185i	\(0.848500\pi\)
\(23\)	−6.44210	+	1.72616i	−1.34327	+	0.359928i	−0.857647	−	0.514239i	\(-0.828074\pi\)
							−0.485624	+	0.874168i	\(0.661408\pi\)
\(29\)			4.72835i			0.878033i	0.898479	+	0.439016i	\(0.144673\pi\)
							−0.898479	+	0.439016i	\(0.855327\pi\)
\(31\)	−1.70170	+	0.982474i	−0.305634	+	0.176458i	−0.644971	−	0.764207i	\(-0.723131\pi\)
							0.339337	+	0.940665i	\(0.389797\pi\)
\(37\)	−7.20177	+	1.92971i	−1.18396	+	0.317242i	−0.796498	−	0.604641i	\(-0.793316\pi\)
							−0.387467	+	0.921884i	\(0.626650\pi\)
\(41\)	4.01882			0.627634			0.313817	−	0.949483i	\(-0.398392\pi\)
							0.313817	+	0.949483i	\(0.398392\pi\)
\(43\)	1.88664	+	1.88664i	0.287710	+	0.287710i	0.836174	−	0.548464i	\(-0.184787\pi\)
							−0.548464	+	0.836174i	\(0.684787\pi\)
\(47\)	−1.70358	−	6.35786i	−0.248493	−	0.927389i	−0.971595	−	0.236648i	\(-0.923951\pi\)
							0.723102	−	0.690741i	\(-0.242716\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	980.2.x.m.667.18		72
4.3	odd	2	inner	980.2.x.m.667.16		72
5.3	odd	4	inner	980.2.x.m.863.10		72
7.2	even	3		980.2.k.j.687.6		36
7.3	odd	6		140.2.w.b.67.8	yes	72
7.4	even	3	inner	980.2.x.m.67.8		72
7.5	odd	6		980.2.k.k.687.6		36
7.6	odd	2		140.2.w.b.107.18	yes	72
20.3	even	4	inner	980.2.x.m.863.8		72
28.3	even	6		140.2.w.b.67.10	yes	72
28.11	odd	6	inner	980.2.x.m.67.10		72
28.19	even	6		980.2.k.k.687.5		36
28.23	odd	6		980.2.k.j.687.5		36
28.27	even	2		140.2.w.b.107.16	yes	72
35.3	even	12		140.2.w.b.123.16	yes	72
35.13	even	4		140.2.w.b.23.10	yes	72
35.17	even	12		700.2.be.e.543.3		72
35.18	odd	12	inner	980.2.x.m.263.16		72
35.23	odd	12		980.2.k.j.883.5		36
35.24	odd	6		700.2.be.e.207.11		72
35.27	even	4		700.2.be.e.443.9		72
35.33	even	12		980.2.k.k.883.5		36
35.34	odd	2		700.2.be.e.107.1		72
140.3	odd	12		140.2.w.b.123.18	yes	72
140.23	even	12		980.2.k.j.883.6		36
140.27	odd	4		700.2.be.e.443.11		72
140.59	even	6		700.2.be.e.207.9		72
140.83	odd	4		140.2.w.b.23.8	✓	72
140.87	odd	12		700.2.be.e.543.1		72
140.103	odd	12		980.2.k.k.883.6		36
140.123	even	12	inner	980.2.x.m.263.18		72
140.139	even	2		700.2.be.e.107.3		72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
140.2.w.b.23.8	✓	72	140.83	odd	4
140.2.w.b.23.10	yes	72	35.13	even	4
140.2.w.b.67.8	yes	72	7.3	odd	6
140.2.w.b.67.10	yes	72	28.3	even	6
140.2.w.b.107.16	yes	72	28.27	even	2
140.2.w.b.107.18	yes	72	7.6	odd	2
140.2.w.b.123.16	yes	72	35.3	even	12
140.2.w.b.123.18	yes	72	140.3	odd	12
700.2.be.e.107.1		72	35.34	odd	2
700.2.be.e.107.3		72	140.139	even	2
700.2.be.e.207.9		72	140.59	even	6
700.2.be.e.207.11		72	35.24	odd	6
700.2.be.e.443.9		72	35.27	even	4
700.2.be.e.443.11		72	140.27	odd	4
700.2.be.e.543.1		72	140.87	odd	12
700.2.be.e.543.3		72	35.17	even	12
980.2.k.j.687.5		36	28.23	odd	6
980.2.k.j.687.6		36	7.2	even	3
980.2.k.j.883.5		36	35.23	odd	12
980.2.k.j.883.6		36	140.23	even	12
980.2.k.k.687.5		36	28.19	even	6
980.2.k.k.687.6		36	7.5	odd	6
980.2.k.k.883.5		36	35.33	even	12
980.2.k.k.883.6		36	140.103	odd	12
980.2.x.m.67.8		72	7.4	even	3	inner
980.2.x.m.67.10		72	28.11	odd	6	inner
980.2.x.m.263.16		72	35.18	odd	12	inner
980.2.x.m.263.18		72	140.123	even	12	inner
980.2.x.m.667.16		72	4.3	odd	2	inner
980.2.x.m.667.18		72	1.1	even	1	trivial
980.2.x.m.863.8		72	20.3	even	4	inner
980.2.x.m.863.10		72	5.3	odd	4	inner