Embedded newform 140.2.w.b.123.7

• Label: 140.2.w.b.123.7
• 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			123.7
Character	\(\chi\)	\(=\)	140.123
Dual form			140.2.w.b.107.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.599707 - 1.28076i) q^{2} +(0.147892 + 0.551941i) q^{3} +(-1.28070 + 1.53616i) q^{4} +(1.15064 + 1.91729i) q^{5} +(0.618214 - 0.520418i) q^{6} +(0.735093 + 2.54158i) q^{7} +(2.73551 + 0.719030i) q^{8} +(2.31531 - 1.33674i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q + 2 q^{2} - 8 q^{5} - 16 q^{6} - 4 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.599707	−	1.28076i	−0.424057	−	0.905636i
							\(3\)	0.147892	+	0.551941i	0.0853856	+	0.318663i	0.995387	−	0.0959419i	\(-0.0305863\pi\)
−0.910001	+	0.414605i	\(0.863920\pi\)
\(5\)	1.15064	+	1.91729i	0.514584	+	0.857440i
							\(7\)	0.735093	+	2.54158i	0.277839	+	0.960628i
\(11\)	−3.42960	−	1.98008i	−1.03406	−	0.597017i								−0.115917	−	0.993259i	\(-0.536981\pi\)
							−0.918146	+	0.396242i	\(0.870314\pi\)
\(13\)	2.04987	+	2.04987i	0.568532	+	0.568532i	0.931717	−	0.363185i	\(-0.118311\pi\)
							−0.363185	+	0.931717i	\(0.618311\pi\)
\(17\)	−0.155552	−	0.580528i	−0.0377269	−	0.140799i	0.944493	−	0.328530i	\(-0.106553\pi\)
							−0.982220	+	0.187731i	\(0.939887\pi\)
\(19\)	−1.00269	−	1.73671i	−0.230033	−	0.398428i	0.727785	−	0.685805i	\(-0.240550\pi\)
							−0.957817	+	0.287377i	\(0.907217\pi\)
\(23\)	2.39119	+	0.640717i	0.498597	+	0.133599i	0.499349	−	0.866401i	\(-0.333573\pi\)
							−0.000751542		1.00000i	\(0.500239\pi\)
\(29\)		−	7.25964i		−	1.34808i	−0.738694	−	0.674041i	\(-0.764557\pi\)
							0.738694	−	0.674041i	\(-0.235443\pi\)
\(31\)	2.83005	+	1.63393i	0.508292	+	0.293462i	0.732131	−	0.681164i	\(-0.238526\pi\)
							−0.223840	+	0.974626i	\(0.571859\pi\)
\(37\)	−9.29977	−	2.49187i	−1.52887	−	0.409660i	−0.606222	−	0.795295i	\(-0.707316\pi\)
							−0.922651	+	0.385635i	\(0.873982\pi\)
\(41\)	−3.35439			−0.523868			−0.261934	−	0.965086i	\(-0.584360\pi\)
							−0.261934	+	0.965086i	\(0.584360\pi\)
\(43\)	−3.31777	+	3.31777i	−0.505955	+	0.505955i	−0.913282	−	0.407327i	\(-0.866461\pi\)
							0.407327	+	0.913282i	\(0.366461\pi\)
\(47\)	3.24611	−	12.1147i	0.473494	−	1.76710i	−0.153572	−	0.988137i	\(-0.549078\pi\)
							0.627066	−	0.778966i	\(-0.284256\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	140.2.w.b.123.7	yes	72
4.3	odd	2	inner	140.2.w.b.123.3	yes	72
5.2	odd	4	inner	140.2.w.b.67.17	yes	72
5.3	odd	4		700.2.be.e.207.2		72
5.4	even	2		700.2.be.e.543.12		72
7.2	even	3	inner	140.2.w.b.23.6	✓	72
7.3	odd	6		980.2.k.j.883.17		36
7.4	even	3		980.2.k.k.883.17		36
7.5	odd	6		980.2.x.m.863.6		72
7.6	odd	2		980.2.x.m.263.7		72
20.3	even	4		700.2.be.e.207.13		72
20.7	even	4	inner	140.2.w.b.67.6	yes	72
20.19	odd	2		700.2.be.e.543.16		72
28.3	even	6		980.2.k.j.883.9		36
28.11	odd	6		980.2.k.k.883.9		36
28.19	even	6		980.2.x.m.863.17		72
28.23	odd	6	inner	140.2.w.b.23.17	yes	72
28.27	even	2		980.2.x.m.263.3		72
35.2	odd	12	inner	140.2.w.b.107.3	yes	72
35.9	even	6		700.2.be.e.443.13		72
35.12	even	12		980.2.x.m.667.3		72
35.17	even	12		980.2.k.j.687.9		36
35.23	odd	12		700.2.be.e.107.16		72
35.27	even	4		980.2.x.m.67.17		72
35.32	odd	12		980.2.k.k.687.9		36
140.23	even	12		700.2.be.e.107.12		72
140.27	odd	4		980.2.x.m.67.6		72
140.47	odd	12		980.2.x.m.667.7		72
140.67	even	12		980.2.k.k.687.17		36
140.79	odd	6		700.2.be.e.443.2		72
140.87	odd	12		980.2.k.j.687.17		36
140.107	even	12	inner	140.2.w.b.107.7	yes	72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
140.2.w.b.23.6	✓	72	7.2	even	3	inner
140.2.w.b.23.17	yes	72	28.23	odd	6	inner
140.2.w.b.67.6	yes	72	20.7	even	4	inner
140.2.w.b.67.17	yes	72	5.2	odd	4	inner
140.2.w.b.107.3	yes	72	35.2	odd	12	inner
140.2.w.b.107.7	yes	72	140.107	even	12	inner
140.2.w.b.123.3	yes	72	4.3	odd	2	inner
140.2.w.b.123.7	yes	72	1.1	even	1	trivial
700.2.be.e.107.12		72	140.23	even	12
700.2.be.e.107.16		72	35.23	odd	12
700.2.be.e.207.2		72	5.3	odd	4
700.2.be.e.207.13		72	20.3	even	4
700.2.be.e.443.2		72	140.79	odd	6
700.2.be.e.443.13		72	35.9	even	6
700.2.be.e.543.12		72	5.4	even	2
700.2.be.e.543.16		72	20.19	odd	2
980.2.k.j.687.9		36	35.17	even	12
980.2.k.j.687.17		36	140.87	odd	12
980.2.k.j.883.9		36	28.3	even	6
980.2.k.j.883.17		36	7.3	odd	6
980.2.k.k.687.9		36	35.32	odd	12
980.2.k.k.687.17		36	140.67	even	12
980.2.k.k.883.9		36	28.11	odd	6
980.2.k.k.883.17		36	7.4	even	3
980.2.x.m.67.6		72	140.27	odd	4
980.2.x.m.67.17		72	35.27	even	4
980.2.x.m.263.3		72	28.27	even	2
980.2.x.m.263.7		72	7.6	odd	2
980.2.x.m.667.3		72	35.12	even	12
980.2.x.m.667.7		72	140.47	odd	12
980.2.x.m.863.6		72	7.5	odd	6
980.2.x.m.863.17		72	28.19	even	6