Embedded newform 140.2.w.b.123.1

• Label: 140.2.w.b.123.1
• Level: $140$
• Weight: $2$
• Character: 140.123
• Analytic conductor: $1.118$
• Analytic rank: $0$
• Dimension: $72$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 140 = 2^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	140.w (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.11790562830\)
Analytic rank:	\(0\)
Dimension:	\(72\)
Relative dimension:	\(18\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			123.1
Character	\(\chi\)	\(=\)	140.123
Dual form			140.2.w.b.107.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.39811 + 0.212826i) q^{2} +(-0.216303 - 0.807254i) q^{3} +(1.90941 - 0.595107i) q^{4} +(-1.42448 - 1.72362i) q^{5} +(0.474220 + 1.08259i) q^{6} +(-2.62434 + 0.335905i) q^{7} +(-2.54291 + 1.23840i) q^{8} +(1.99320 - 1.15078i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q + 2 q^{2} - 8 q^{5} - 16 q^{6} - 4 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/140\mathbb{Z}\right)^\times\).

\(n\)	\(57\)	\(71\)	\(101\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(-1\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.39811	+	0.212826i	−0.988611	+	0.150491i
							\(3\)	−0.216303	−	0.807254i	−0.124883	−	0.466069i	0.874953	−	0.484208i	\(-0.160892\pi\)
−0.999835	+	0.0181397i	\(0.994226\pi\)
\(5\)	−1.42448	−	1.72362i	−0.637048	−	0.770824i
							\(7\)	−2.62434	+	0.335905i	−0.991908	+	0.126960i
\(11\)	−4.21811	−	2.43533i	−1.27181	−	0.734279i								−0.296480	−	0.955039i	\(-0.595813\pi\)
							−0.975328	+	0.220760i	\(0.929146\pi\)
\(13\)	0.247904	+	0.247904i	0.0687562	+	0.0687562i	0.740649	−	0.671892i	\(-0.234518\pi\)
							−0.671892	+	0.740649i	\(0.734518\pi\)
\(17\)	−1.56358	−	5.83537i	−0.379224	−	1.41528i	−0.847073	−	0.531476i	\(-0.821638\pi\)
							0.467849	−	0.883808i	\(-0.345029\pi\)
\(19\)	−1.48858	−	2.57830i	−0.341505	−	0.591503i	0.643208	−	0.765692i	\(-0.277603\pi\)
							−0.984712	+	0.174188i	\(0.944270\pi\)
\(23\)	6.36754	+	1.70618i	1.32772	+	0.355762i	0.851866	−	0.523760i	\(-0.175471\pi\)
							0.475858	+	0.879522i	\(0.342138\pi\)
\(29\)			4.67111i			0.867403i	0.901057	+	0.433701i	\(0.142793\pi\)
							−0.901057	+	0.433701i	\(0.857207\pi\)
\(31\)	5.75848	+	3.32466i	1.03425	+	0.597127i	0.918200	−	0.396116i	\(-0.129642\pi\)
							0.116053	+	0.993243i	\(0.462976\pi\)
\(37\)	2.39496	+	0.641729i	0.393730	+	0.105500i	0.450251	−	0.892902i	\(-0.351334\pi\)
							−0.0565217	+	0.998401i	\(0.518001\pi\)
\(41\)	−1.15345			−0.180138			−0.0900692	−	0.995936i	\(-0.528709\pi\)
							−0.0900692	+	0.995936i	\(0.528709\pi\)
\(43\)	3.23212	−	3.23212i	0.492894	−	0.492894i	−0.416323	−	0.909217i	\(-0.636682\pi\)
							0.909217	+	0.416323i	\(0.136682\pi\)
\(47\)	1.27127	−	4.74443i	0.185433	−	0.692047i	−0.809104	−	0.587666i	\(-0.800047\pi\)
							0.994537	−	0.104381i	\(-0.0332862\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	140.2.w.b.123.1	yes	72
4.3	odd	2	inner	140.2.w.b.123.4	yes	72
5.2	odd	4	inner	140.2.w.b.67.9	yes	72
5.3	odd	4		700.2.be.e.207.10		72
5.4	even	2		700.2.be.e.543.18		72
7.2	even	3	inner	140.2.w.b.23.12	yes	72
7.3	odd	6		980.2.k.j.883.12		36
7.4	even	3		980.2.k.k.883.12		36
7.5	odd	6		980.2.x.m.863.12		72
7.6	odd	2		980.2.x.m.263.1		72
20.3	even	4		700.2.be.e.207.7		72
20.7	even	4	inner	140.2.w.b.67.12	yes	72
20.19	odd	2		700.2.be.e.543.15		72
28.3	even	6		980.2.k.j.883.16		36
28.11	odd	6		980.2.k.k.883.16		36
28.19	even	6		980.2.x.m.863.9		72
28.23	odd	6	inner	140.2.w.b.23.9	✓	72
28.27	even	2		980.2.x.m.263.4		72
35.2	odd	12	inner	140.2.w.b.107.4	yes	72
35.9	even	6		700.2.be.e.443.7		72
35.12	even	12		980.2.x.m.667.4		72
35.17	even	12		980.2.k.j.687.16		36
35.23	odd	12		700.2.be.e.107.15		72
35.27	even	4		980.2.x.m.67.9		72
35.32	odd	12		980.2.k.k.687.16		36
140.23	even	12		700.2.be.e.107.18		72
140.27	odd	4		980.2.x.m.67.12		72
140.47	odd	12		980.2.x.m.667.1		72
140.67	even	12		980.2.k.k.687.12		36
140.79	odd	6		700.2.be.e.443.10		72
140.87	odd	12		980.2.k.j.687.12		36
140.107	even	12	inner	140.2.w.b.107.1	yes	72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
140.2.w.b.23.9	✓	72	28.23	odd	6	inner
140.2.w.b.23.12	yes	72	7.2	even	3	inner
140.2.w.b.67.9	yes	72	5.2	odd	4	inner
140.2.w.b.67.12	yes	72	20.7	even	4	inner
140.2.w.b.107.1	yes	72	140.107	even	12	inner
140.2.w.b.107.4	yes	72	35.2	odd	12	inner
140.2.w.b.123.1	yes	72	1.1	even	1	trivial
140.2.w.b.123.4	yes	72	4.3	odd	2	inner
700.2.be.e.107.15		72	35.23	odd	12
700.2.be.e.107.18		72	140.23	even	12
700.2.be.e.207.7		72	20.3	even	4
700.2.be.e.207.10		72	5.3	odd	4
700.2.be.e.443.7		72	35.9	even	6
700.2.be.e.443.10		72	140.79	odd	6
700.2.be.e.543.15		72	20.19	odd	2
700.2.be.e.543.18		72	5.4	even	2
980.2.k.j.687.12		36	140.87	odd	12
980.2.k.j.687.16		36	35.17	even	12
980.2.k.j.883.12		36	7.3	odd	6
980.2.k.j.883.16		36	28.3	even	6
980.2.k.k.687.12		36	140.67	even	12
980.2.k.k.687.16		36	35.32	odd	12
980.2.k.k.883.12		36	7.4	even	3
980.2.k.k.883.16		36	28.11	odd	6
980.2.x.m.67.9		72	35.27	even	4
980.2.x.m.67.12		72	140.27	odd	4
980.2.x.m.263.1		72	7.6	odd	2
980.2.x.m.263.4		72	28.27	even	2
980.2.x.m.667.1		72	140.47	odd	12
980.2.x.m.667.4		72	35.12	even	12
980.2.x.m.863.9		72	28.19	even	6
980.2.x.m.863.12		72	7.5	odd	6