Embedded newform 140.2.w.b.123.13

• Label: 140.2.w.b.123.13
• 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			123.13
Character	\(\chi\)	\(=\)	140.123
Dual form			140.2.w.b.107.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.839014 - 1.13844i) q^{2} +(-0.290169 - 1.08292i) q^{3} +(-0.592112 - 1.91034i) q^{4} +(-2.11448 + 0.727309i) q^{5} +(-1.47630 - 0.578247i) q^{6} +(2.16744 - 1.51730i) q^{7} +(-2.67161 - 0.928715i) q^{8} +(1.50955 - 0.871538i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q + 2 q^{2} - 8 q^{5} - 16 q^{6} - 4 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.839014	−	1.13844i	0.593272	−	0.805002i
							\(3\)	−0.290169	−	1.08292i	−0.167529	−	0.625227i	−0.997704	−	0.0677240i	\(-0.978426\pi\)
0.830175	−	0.557503i	\(-0.188240\pi\)
\(5\)	−2.11448	+	0.727309i	−0.945624	+	0.325262i
							\(7\)	2.16744	−	1.51730i	0.819217	−	0.573484i
\(11\)	−2.58391	−	1.49182i	−0.779077	−	0.449800i								0.0570261	−	0.998373i	\(-0.481838\pi\)
							−0.836103	+	0.548572i	\(0.815172\pi\)
\(13\)	4.05418	+	4.05418i	1.12443	+	1.12443i	0.991068	+	0.133359i	\(0.0425763\pi\)
							0.133359	+	0.991068i	\(0.457424\pi\)
\(17\)	0.617565	+	2.30478i	0.149782	+	0.558992i	0.999496	+	0.0317490i	\(0.0101077\pi\)
							−0.849714	+	0.527243i	\(0.823226\pi\)
\(19\)	1.25694	+	2.17708i	0.288362	+	0.499457i	0.973419	−	0.229032i	\(-0.0735561\pi\)
							−0.685057	+	0.728489i	\(0.740223\pi\)
\(23\)	5.79414	+	1.55254i	1.20816	+	0.323726i	0.806041	−	0.591860i	\(-0.201606\pi\)
							0.402122	+	0.915586i	\(0.368273\pi\)
\(29\)		−	2.55433i		−	0.474327i	−0.971470	−	0.237163i	\(-0.923782\pi\)
							0.971470	−	0.237163i	\(-0.0762177\pi\)
\(31\)	−3.12248	−	1.80277i	−0.560815	−	0.323787i	0.192658	−	0.981266i	\(-0.438289\pi\)
							−0.753472	+	0.657479i	\(0.771623\pi\)
\(37\)	4.61014	+	1.23528i	0.757903	+	0.203079i	0.617021	−	0.786947i	\(-0.288339\pi\)
							0.140882	+	0.990026i	\(0.455006\pi\)
\(41\)	−7.93727			−1.23959			−0.619797	−	0.784763i	\(-0.712785\pi\)
							−0.619797	+	0.784763i	\(0.712785\pi\)
\(43\)	−7.62646	+	7.62646i	−1.16302	+	1.16302i	−0.179215	+	0.983810i	\(0.557356\pi\)
							−0.983810	+	0.179215i	\(0.942644\pi\)
\(47\)	−0.765987	+	2.85870i	−0.111731	+	0.416985i	−0.999022	−	0.0442256i	\(-0.985918\pi\)
							0.887291	+	0.461210i	\(0.152585\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	140.2.w.b.123.13	yes	72
4.3	odd	2	inner	140.2.w.b.123.10	yes	72
5.2	odd	4	inner	140.2.w.b.67.14	yes	72
5.3	odd	4		700.2.be.e.207.5		72
5.4	even	2		700.2.be.e.543.6		72
7.2	even	3	inner	140.2.w.b.23.2	✓	72
7.3	odd	6		980.2.k.j.883.13		36
7.4	even	3		980.2.k.k.883.13		36
7.5	odd	6		980.2.x.m.863.2		72
7.6	odd	2		980.2.x.m.263.13		72
20.3	even	4		700.2.be.e.207.17		72
20.7	even	4	inner	140.2.w.b.67.2	yes	72
20.19	odd	2		700.2.be.e.543.9		72
28.3	even	6		980.2.k.j.883.3		36
28.11	odd	6		980.2.k.k.883.3		36
28.19	even	6		980.2.x.m.863.14		72
28.23	odd	6	inner	140.2.w.b.23.14	yes	72
28.27	even	2		980.2.x.m.263.10		72
35.2	odd	12	inner	140.2.w.b.107.10	yes	72
35.9	even	6		700.2.be.e.443.17		72
35.12	even	12		980.2.x.m.667.10		72
35.17	even	12		980.2.k.j.687.3		36
35.23	odd	12		700.2.be.e.107.9		72
35.27	even	4		980.2.x.m.67.14		72
35.32	odd	12		980.2.k.k.687.3		36
140.23	even	12		700.2.be.e.107.6		72
140.27	odd	4		980.2.x.m.67.2		72
140.47	odd	12		980.2.x.m.667.13		72
140.67	even	12		980.2.k.k.687.13		36
140.79	odd	6		700.2.be.e.443.5		72
140.87	odd	12		980.2.k.j.687.13		36
140.107	even	12	inner	140.2.w.b.107.13	yes	72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
140.2.w.b.23.2	✓	72	7.2	even	3	inner
140.2.w.b.23.14	yes	72	28.23	odd	6	inner
140.2.w.b.67.2	yes	72	20.7	even	4	inner
140.2.w.b.67.14	yes	72	5.2	odd	4	inner
140.2.w.b.107.10	yes	72	35.2	odd	12	inner
140.2.w.b.107.13	yes	72	140.107	even	12	inner
140.2.w.b.123.10	yes	72	4.3	odd	2	inner
140.2.w.b.123.13	yes	72	1.1	even	1	trivial
700.2.be.e.107.6		72	140.23	even	12
700.2.be.e.107.9		72	35.23	odd	12
700.2.be.e.207.5		72	5.3	odd	4
700.2.be.e.207.17		72	20.3	even	4
700.2.be.e.443.5		72	140.79	odd	6
700.2.be.e.443.17		72	35.9	even	6
700.2.be.e.543.6		72	5.4	even	2
700.2.be.e.543.9		72	20.19	odd	2
980.2.k.j.687.3		36	35.17	even	12
980.2.k.j.687.13		36	140.87	odd	12
980.2.k.j.883.3		36	28.3	even	6
980.2.k.j.883.13		36	7.3	odd	6
980.2.k.k.687.3		36	35.32	odd	12
980.2.k.k.687.13		36	140.67	even	12
980.2.k.k.883.3		36	28.11	odd	6
980.2.k.k.883.13		36	7.4	even	3
980.2.x.m.67.2		72	140.27	odd	4
980.2.x.m.67.14		72	35.27	even	4
980.2.x.m.263.10		72	28.27	even	2
980.2.x.m.263.13		72	7.6	odd	2
980.2.x.m.667.10		72	35.12	even	12
980.2.x.m.667.13		72	140.47	odd	12
980.2.x.m.863.2		72	7.5	odd	6
980.2.x.m.863.14		72	28.19	even	6