Embedded newform 140.2.w.b.107.1

• Label: 140.2.w.b.107.1
• Level: $140$
• Weight: $2$
• Character: 140.107
• 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			107.1
Character	\(\chi\)	\(=\)	140.107
Dual form			140.2.w.b.123.1

$q$-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 \(\left(\mathbb{Z}/140\mathbb{Z}\right)^\times\).

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

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	−0.999835	−	0.0181397i	\(-0.994226\pi\)
0.874953	+	0.484208i	\(0.160892\pi\)
\(5\)	−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								−0.975328	−	0.220760i	\(-0.929146\pi\)
							−0.296480	+	0.955039i	\(0.595813\pi\)
\(13\)	0.247904	−	0.247904i	0.0687562	−	0.0687562i	−0.671892	−	0.740649i	\(-0.734518\pi\)
							0.740649	+	0.671892i	\(0.234518\pi\)
\(17\)	−1.56358	+	5.83537i	−0.379224	+	1.41528i	0.467849	+	0.883808i	\(0.345029\pi\)
							−0.847073	+	0.531476i	\(0.821638\pi\)
\(19\)	−1.48858	+	2.57830i	−0.341505	+	0.591503i	−0.984712	−	0.174188i	\(-0.944270\pi\)
							0.643208	+	0.765692i	\(0.277603\pi\)
\(23\)	6.36754	−	1.70618i	1.32772	−	0.355762i	0.475858	−	0.879522i	\(-0.342138\pi\)
							0.851866	+	0.523760i	\(0.175471\pi\)
\(29\)		−	4.67111i		−	0.867403i	−0.901057	−	0.433701i	\(-0.857207\pi\)
							0.901057	−	0.433701i	\(-0.142793\pi\)
\(31\)	5.75848	−	3.32466i	1.03425	−	0.597127i	0.116053	−	0.993243i	\(-0.462976\pi\)
							0.918200	+	0.396116i	\(0.129642\pi\)
\(37\)	2.39496	−	0.641729i	0.393730	−	0.105500i	−0.0565217	−	0.998401i	\(-0.518001\pi\)
							0.450251	+	0.892902i	\(0.351334\pi\)
\(41\)	−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	0.909217	−	0.416323i	\(-0.136682\pi\)
							−0.416323	+	0.909217i	\(0.636682\pi\)
\(47\)	1.27127	+	4.74443i	0.185433	+	0.692047i	0.994537	+	0.104381i	\(0.0332862\pi\)
							−0.809104	+	0.587666i	\(0.800047\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
140.2.w.b.23.9	✓	72	5.3	odd	4	inner
140.2.w.b.23.12	yes	72	20.3	even	4	inner
140.2.w.b.67.9	yes	72	28.11	odd	6	inner
140.2.w.b.67.12	yes	72	7.4	even	3	inner
140.2.w.b.107.1	yes	72	1.1	even	1	trivial
140.2.w.b.107.4	yes	72	4.3	odd	2	inner
140.2.w.b.123.1	yes	72	140.123	even	12	inner
140.2.w.b.123.4	yes	72	35.18	odd	12	inner
700.2.be.e.107.15		72	20.19	odd	2
700.2.be.e.107.18		72	5.4	even	2
700.2.be.e.207.7		72	35.4	even	6
700.2.be.e.207.10		72	140.39	odd	6
700.2.be.e.443.7		72	20.7	even	4
700.2.be.e.443.10		72	5.2	odd	4
700.2.be.e.543.15		72	35.32	odd	12
700.2.be.e.543.18		72	140.67	even	12
980.2.k.j.687.12		36	7.5	odd	6
980.2.k.j.687.16		36	28.19	even	6
980.2.k.j.883.12		36	140.103	odd	12
980.2.k.j.883.16		36	35.33	even	12
980.2.k.k.687.12		36	7.2	even	3
980.2.k.k.687.16		36	28.23	odd	6
980.2.k.k.883.12		36	140.23	even	12
980.2.k.k.883.16		36	35.23	odd	12
980.2.x.m.67.9		72	28.3	even	6
980.2.x.m.67.12		72	7.3	odd	6
980.2.x.m.263.1		72	140.3	odd	12
980.2.x.m.263.4		72	35.3	even	12
980.2.x.m.667.1		72	7.6	odd	2
980.2.x.m.667.4		72	28.27	even	2
980.2.x.m.863.9		72	35.13	even	4
980.2.x.m.863.12		72	140.83	odd	4