Embedded newform 140.2.w.b.123.9

• Label: 140.2.w.b.123.9
• Level: $140$
• Weight: $2$
• Character: 140.123
• Analytic conductor: $1.118$
• Analytic rank: $0$
• Dimension: $72$
• 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			123.9
Character	\(\chi\)	\(=\)	140.123
Dual form			140.2.w.b.107.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0671791 + 1.41262i) q^{2} +(0.638980 + 2.38471i) q^{3} +(-1.99097 + 0.189797i) q^{4} +(2.14503 + 0.631526i) q^{5} +(-3.32575 + 1.06284i) q^{6} +(1.60796 - 2.10106i) q^{7} +(-0.401862 - 2.79973i) q^{8} +(-2.68045 + 1.54756i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q + 2 q^{2} - 8 q^{5} - 16 q^{6} - 4 q^{8}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0671791	+	1.41262i	0.0475028	+	0.998871i
							\(3\)	0.638980	+	2.38471i	0.368915	+	1.37681i	0.862035	+	0.506849i	\(0.169190\pi\)
−0.493119	+	0.869962i	\(0.664143\pi\)
\(5\)	2.14503	+	0.631526i	0.959289	+	0.282427i
							\(7\)	1.60796	−	2.10106i	0.607751	−	0.794128i
\(11\)	−4.09588	−	2.36476i	−1.23496	−	0.713002i								−0.266897	−	0.963725i	\(-0.585998\pi\)
							−0.968059	+	0.250723i	\(0.919332\pi\)
\(13\)	0.0592655	+	0.0592655i	0.0164373	+	0.0164373i	0.715278	−	0.698840i	\(-0.246300\pi\)
							−0.698840	+	0.715278i	\(0.746300\pi\)
\(17\)	−1.27942	−	4.77484i	−0.310304	−	1.15807i	−0.928283	−	0.371875i	\(-0.878715\pi\)
							0.617979	−	0.786195i	\(-0.287952\pi\)
\(19\)	1.31544	+	2.27842i	0.301784	+	0.522705i	0.976540	−	0.215336i	\(-0.0690846\pi\)
							−0.674756	+	0.738041i	\(0.735751\pi\)
\(23\)	−1.09265	−	0.292774i	−0.227832	−	0.0610475i	0.143097	−	0.989709i	\(-0.454294\pi\)
							−0.370929	+	0.928661i	\(0.620961\pi\)
\(29\)			7.27332i			1.35062i	0.737533	+	0.675311i	\(0.235991\pi\)
							−0.737533	+	0.675311i	\(0.764009\pi\)
\(31\)	−4.01558	−	2.31840i	−0.721219	−	0.416396i	0.0939821	−	0.995574i	\(-0.470040\pi\)
							−0.815201	+	0.579178i	\(0.803374\pi\)
\(37\)	2.22779	+	0.596933i	0.366246	+	0.0981352i	0.437248	−	0.899341i	\(-0.355953\pi\)
							−0.0710024	+	0.997476i	\(0.522620\pi\)
\(41\)	−5.71767			−0.892950			−0.446475	−	0.894796i	\(-0.647321\pi\)
							−0.446475	+	0.894796i	\(0.647321\pi\)
\(43\)	1.57302	−	1.57302i	0.239883	−	0.239883i	−0.576918	−	0.816802i	\(-0.695745\pi\)
							0.816802	+	0.576918i	\(0.195745\pi\)
\(47\)	0.716679	−	2.67468i	0.104538	−	0.390143i	−0.893754	−	0.448558i	\(-0.851938\pi\)
							0.998292	+	0.0584148i	\(0.0186046\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	140.2.w.b.123.9	yes	72
4.3	odd	2	inner	140.2.w.b.123.12	yes	72
5.2	odd	4	inner	140.2.w.b.67.1	yes	72
5.3	odd	4		700.2.be.e.207.18		72
5.4	even	2		700.2.be.e.543.10		72
7.2	even	3	inner	140.2.w.b.23.15	yes	72
7.3	odd	6		980.2.k.j.883.4		36
7.4	even	3		980.2.k.k.883.4		36
7.5	odd	6		980.2.x.m.863.15		72
7.6	odd	2		980.2.x.m.263.9		72
20.3	even	4		700.2.be.e.207.4		72
20.7	even	4	inner	140.2.w.b.67.15	yes	72
20.19	odd	2		700.2.be.e.543.7		72
28.3	even	6		980.2.k.j.883.14		36
28.11	odd	6		980.2.k.k.883.14		36
28.19	even	6		980.2.x.m.863.1		72
28.23	odd	6	inner	140.2.w.b.23.1	✓	72
28.27	even	2		980.2.x.m.263.12		72
35.2	odd	12	inner	140.2.w.b.107.12	yes	72
35.9	even	6		700.2.be.e.443.4		72
35.12	even	12		980.2.x.m.667.12		72
35.17	even	12		980.2.k.j.687.14		36
35.23	odd	12		700.2.be.e.107.7		72
35.27	even	4		980.2.x.m.67.1		72
35.32	odd	12		980.2.k.k.687.14		36
140.23	even	12		700.2.be.e.107.10		72
140.27	odd	4		980.2.x.m.67.15		72
140.47	odd	12		980.2.x.m.667.9		72
140.67	even	12		980.2.k.k.687.4		36
140.79	odd	6		700.2.be.e.443.18		72
140.87	odd	12		980.2.k.j.687.4		36
140.107	even	12	inner	140.2.w.b.107.9	yes	72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
140.2.w.b.23.1	✓	72	28.23	odd	6	inner
140.2.w.b.23.15	yes	72	7.2	even	3	inner
140.2.w.b.67.1	yes	72	5.2	odd	4	inner
140.2.w.b.67.15	yes	72	20.7	even	4	inner
140.2.w.b.107.9	yes	72	140.107	even	12	inner
140.2.w.b.107.12	yes	72	35.2	odd	12	inner
140.2.w.b.123.9	yes	72	1.1	even	1	trivial
140.2.w.b.123.12	yes	72	4.3	odd	2	inner
700.2.be.e.107.7		72	35.23	odd	12
700.2.be.e.107.10		72	140.23	even	12
700.2.be.e.207.4		72	20.3	even	4
700.2.be.e.207.18		72	5.3	odd	4
700.2.be.e.443.4		72	35.9	even	6
700.2.be.e.443.18		72	140.79	odd	6
700.2.be.e.543.7		72	20.19	odd	2
700.2.be.e.543.10		72	5.4	even	2
980.2.k.j.687.4		36	140.87	odd	12
980.2.k.j.687.14		36	35.17	even	12
980.2.k.j.883.4		36	7.3	odd	6
980.2.k.j.883.14		36	28.3	even	6
980.2.k.k.687.4		36	140.67	even	12
980.2.k.k.687.14		36	35.32	odd	12
980.2.k.k.883.4		36	7.4	even	3
980.2.k.k.883.14		36	28.11	odd	6
980.2.x.m.67.1		72	35.27	even	4
980.2.x.m.67.15		72	140.27	odd	4
980.2.x.m.263.9		72	7.6	odd	2
980.2.x.m.263.12		72	28.27	even	2
980.2.x.m.667.9		72	140.47	odd	12
980.2.x.m.667.12		72	35.12	even	12
980.2.x.m.863.1		72	28.19	even	6
980.2.x.m.863.15		72	7.5	odd	6