Embedded newform 980.2.x.m.67.18

• Label: 980.2.x.m.67.18
• Level: $980$
• Weight: $2$
• Character: 980.67
• 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			67.18
Character	\(\chi\)	\(=\)	980.67
Dual form			980.2.x.m.863.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.41374 + 0.0366689i) q^{2} +(-0.402205 + 0.107770i) q^{3} +(1.99731 + 0.103680i) q^{4} +(2.22809 - 0.188711i) q^{5} +(-0.572564 + 0.137611i) q^{6} +(2.81987 + 0.219816i) q^{8} +(-2.44792 + 1.41331i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q + 2 q^{2} + 8 q^{5} + 16 q^{6} - 4 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.41374	+	0.0366689i	0.999664	+	0.0259288i
							\(3\)	−0.402205	+	0.107770i	−0.232213	+	0.0622213i	−0.373049	−	0.927812i	\(-0.621688\pi\)
0.140836	+	0.990033i	\(0.455021\pi\)
\(5\)	2.22809	−	0.188711i	0.996432	−	0.0843941i
							\(7\)		0			0
\(11\)	−0.725638	−	0.418947i	−0.218788	−	0.126317i								0.386601	−	0.922247i	\(-0.373649\pi\)
							−0.605389	+	0.795930i	\(0.706982\pi\)
\(13\)	1.16367	−	1.16367i	0.322743	−	0.322743i	−0.527075	−	0.849819i	\(-0.676711\pi\)
							0.849819	+	0.527075i	\(0.176711\pi\)
\(17\)	4.94841	−	1.32592i	1.20016	−	0.321583i	0.397269	−	0.917702i	\(-0.369958\pi\)
							0.802896	+	0.596119i	\(0.203291\pi\)
\(19\)	2.91741	+	5.05309i	0.669299	+	1.15926i	0.978101	+	0.208133i	\(0.0667387\pi\)
							−0.308802	+	0.951126i	\(0.599928\pi\)
\(23\)	−0.896861	+	3.34713i	−0.187008	+	0.697925i	0.807183	+	0.590301i	\(0.200991\pi\)
							−0.994192	+	0.107624i	\(0.965676\pi\)
\(29\)		−	5.00172i		−	0.928795i	−0.885627	−	0.464398i	\(-0.846271\pi\)
							0.885627	−	0.464398i	\(-0.153729\pi\)
\(31\)	−7.03267	−	4.06031i	−1.26310	−	0.729254i	−0.289430	−	0.957199i	\(-0.593466\pi\)
							−0.973674	+	0.227945i	\(0.926799\pi\)
\(37\)	−0.190621	+	0.711408i	−0.0313379	+	0.116955i	−0.979823	−	0.199867i	\(-0.935949\pi\)
							0.948485	+	0.316822i	\(0.102616\pi\)
\(41\)	−0.0958388			−0.0149675			−0.00748375	−	0.999972i	\(-0.502382\pi\)
							−0.00748375	+	0.999972i	\(0.502382\pi\)
\(43\)	−4.87975	−	4.87975i	−0.744155	−	0.744155i	0.229220	−	0.973375i	\(-0.426383\pi\)
							−0.973375	+	0.229220i	\(0.926383\pi\)
\(47\)	−5.28468	−	1.41603i	−0.770850	−	0.206549i	−0.148103	−	0.988972i	\(-0.547317\pi\)
							−0.622747	+	0.782423i	\(0.713983\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	980.2.x.m.67.18		72
4.3	odd	2	inner	980.2.x.m.67.4		72
5.3	odd	4	inner	980.2.x.m.263.8		72
7.2	even	3	inner	980.2.x.m.667.6		72
7.3	odd	6		980.2.k.k.687.7		36
7.4	even	3		980.2.k.j.687.7		36
7.5	odd	6		140.2.w.b.107.6	yes	72
7.6	odd	2		140.2.w.b.67.18	yes	72
20.3	even	4	inner	980.2.x.m.263.6		72
28.3	even	6		980.2.k.k.687.15		36
28.11	odd	6		980.2.k.j.687.15		36
28.19	even	6		140.2.w.b.107.8	yes	72
28.23	odd	6	inner	980.2.x.m.667.8		72
28.27	even	2		140.2.w.b.67.4	yes	72
35.3	even	12		980.2.k.k.883.15		36
35.12	even	12		700.2.be.e.443.15		72
35.13	even	4		140.2.w.b.123.8	yes	72
35.18	odd	12		980.2.k.j.883.15		36
35.19	odd	6		700.2.be.e.107.13		72
35.23	odd	12	inner	980.2.x.m.863.4		72
35.27	even	4		700.2.be.e.543.11		72
35.33	even	12		140.2.w.b.23.4	✓	72
35.34	odd	2		700.2.be.e.207.1		72
140.3	odd	12		980.2.k.k.883.7		36
140.19	even	6		700.2.be.e.107.11		72
140.23	even	12	inner	980.2.x.m.863.18		72
140.27	odd	4		700.2.be.e.543.13		72
140.47	odd	12		700.2.be.e.443.1		72
140.83	odd	4		140.2.w.b.123.6	yes	72
140.103	odd	12		140.2.w.b.23.18	yes	72
140.123	even	12		980.2.k.j.883.7		36
140.139	even	2		700.2.be.e.207.15		72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
140.2.w.b.23.4	✓	72	35.33	even	12
140.2.w.b.23.18	yes	72	140.103	odd	12
140.2.w.b.67.4	yes	72	28.27	even	2
140.2.w.b.67.18	yes	72	7.6	odd	2
140.2.w.b.107.6	yes	72	7.5	odd	6
140.2.w.b.107.8	yes	72	28.19	even	6
140.2.w.b.123.6	yes	72	140.83	odd	4
140.2.w.b.123.8	yes	72	35.13	even	4
700.2.be.e.107.11		72	140.19	even	6
700.2.be.e.107.13		72	35.19	odd	6
700.2.be.e.207.1		72	35.34	odd	2
700.2.be.e.207.15		72	140.139	even	2
700.2.be.e.443.1		72	140.47	odd	12
700.2.be.e.443.15		72	35.12	even	12
700.2.be.e.543.11		72	35.27	even	4
700.2.be.e.543.13		72	140.27	odd	4
980.2.k.j.687.7		36	7.4	even	3
980.2.k.j.687.15		36	28.11	odd	6
980.2.k.j.883.7		36	140.123	even	12
980.2.k.j.883.15		36	35.18	odd	12
980.2.k.k.687.7		36	7.3	odd	6
980.2.k.k.687.15		36	28.3	even	6
980.2.k.k.883.7		36	140.3	odd	12
980.2.k.k.883.15		36	35.3	even	12
980.2.x.m.67.4		72	4.3	odd	2	inner
980.2.x.m.67.18		72	1.1	even	1	trivial
980.2.x.m.263.6		72	20.3	even	4	inner
980.2.x.m.263.8		72	5.3	odd	4	inner
980.2.x.m.667.6		72	7.2	even	3	inner
980.2.x.m.667.8		72	28.23	odd	6	inner
980.2.x.m.863.4		72	35.23	odd	12	inner
980.2.x.m.863.18		72	140.23	even	12	inner