Embedded newform 140.2.w.b.123.14

• Label: 140.2.w.b.123.14
• 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			123.14
Character	\(\chi\)	\(=\)	140.123
Dual form			140.2.w.b.107.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.942570 + 1.05431i) q^{2} +(-0.322645 - 1.20413i) q^{3} +(-0.223125 + 1.98751i) q^{4} +(-0.381497 + 2.20328i) q^{5} +(0.965404 - 1.47514i) q^{6} +(2.60693 + 0.451584i) q^{7} +(-2.30576 + 1.63813i) q^{8} +(1.25225 - 0.722989i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q + 2 q^{2} - 8 q^{5} - 16 q^{6} - 4 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.942570	+	1.05431i	0.666497	+	0.745507i
							\(3\)	−0.322645	−	1.20413i	−0.186279	−	0.695203i	−0.994353	−	0.106122i	\(-0.966157\pi\)
0.808074	−	0.589081i	\(-0.200510\pi\)
\(5\)	−0.381497	+	2.20328i	−0.170610	+	0.985339i
							\(7\)	2.60693	+	0.451584i	0.985326	+	0.170683i
\(11\)	0.824629	+	0.476100i	0.248635	+	0.143549i								0.619139	−	0.785281i	\(-0.287482\pi\)
							−0.370504	+	0.928831i	\(0.620815\pi\)
\(13\)	−3.15680	−	3.15680i	−0.875540	−	0.875540i	0.117529	−	0.993069i	\(-0.462503\pi\)
							−0.993069	+	0.117529i	\(0.962503\pi\)
\(17\)	−0.688322	−	2.56885i	−0.166943	−	0.623038i	−0.997784	−	0.0665295i	\(-0.978807\pi\)
							0.830842	−	0.556508i	\(-0.187859\pi\)
\(19\)	−1.99048	−	3.44760i	−0.456646	−	0.790935i	0.542135	−	0.840292i	\(-0.317616\pi\)
							−0.998781	+	0.0493567i	\(0.984283\pi\)
\(23\)	−0.528392	−	0.141582i	−0.110177	−	0.0295219i	0.203309	−	0.979115i	\(-0.434830\pi\)
							−0.313486	+	0.949593i	\(0.601497\pi\)
\(29\)			7.23080i			1.34273i	0.741129	+	0.671363i	\(0.234291\pi\)
							−0.741129	+	0.671363i	\(0.765709\pi\)
\(31\)	3.41863	+	1.97375i	0.614004	+	0.354495i	0.774531	−	0.632536i	\(-0.217986\pi\)
							−0.160527	+	0.987031i	\(0.551319\pi\)
\(37\)	−1.00460	−	0.269180i	−0.165154	−	0.0442530i	0.175294	−	0.984516i	\(-0.443912\pi\)
							−0.340449	+	0.940263i	\(0.610579\pi\)
\(41\)	3.71449			0.580106			0.290053	−	0.957011i	\(-0.406327\pi\)
							0.290053	+	0.957011i	\(0.406327\pi\)
\(43\)	8.73324	−	8.73324i	1.33181	−	1.33181i	0.428052	−	0.903754i	\(-0.359200\pi\)
							0.903754	−	0.428052i	\(-0.140800\pi\)
\(47\)	0.830089	−	3.09794i	0.121081	−	0.451880i	−0.878589	−	0.477579i	\(-0.841514\pi\)
							0.999670	+	0.0256988i	\(0.00818107\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	140.2.w.b.123.14	yes	72
4.3	odd	2	inner	140.2.w.b.123.17	yes	72
5.2	odd	4	inner	140.2.w.b.67.5	yes	72
5.3	odd	4		700.2.be.e.207.14		72
5.4	even	2		700.2.be.e.543.5		72
7.2	even	3	inner	140.2.w.b.23.11	yes	72
7.3	odd	6		980.2.k.j.883.1		36
7.4	even	3		980.2.k.k.883.1		36
7.5	odd	6		980.2.x.m.863.11		72
7.6	odd	2		980.2.x.m.263.14		72
20.3	even	4		700.2.be.e.207.8		72
20.7	even	4	inner	140.2.w.b.67.11	yes	72
20.19	odd	2		700.2.be.e.543.2		72
28.3	even	6		980.2.k.j.883.8		36
28.11	odd	6		980.2.k.k.883.8		36
28.19	even	6		980.2.x.m.863.5		72
28.23	odd	6	inner	140.2.w.b.23.5	✓	72
28.27	even	2		980.2.x.m.263.17		72
35.2	odd	12	inner	140.2.w.b.107.17	yes	72
35.9	even	6		700.2.be.e.443.8		72
35.12	even	12		980.2.x.m.667.17		72
35.17	even	12		980.2.k.j.687.8		36
35.23	odd	12		700.2.be.e.107.2		72
35.27	even	4		980.2.x.m.67.5		72
35.32	odd	12		980.2.k.k.687.8		36
140.23	even	12		700.2.be.e.107.5		72
140.27	odd	4		980.2.x.m.67.11		72
140.47	odd	12		980.2.x.m.667.14		72
140.67	even	12		980.2.k.k.687.1		36
140.79	odd	6		700.2.be.e.443.14		72
140.87	odd	12		980.2.k.j.687.1		36
140.107	even	12	inner	140.2.w.b.107.14	yes	72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
140.2.w.b.23.5	✓	72	28.23	odd	6	inner
140.2.w.b.23.11	yes	72	7.2	even	3	inner
140.2.w.b.67.5	yes	72	5.2	odd	4	inner
140.2.w.b.67.11	yes	72	20.7	even	4	inner
140.2.w.b.107.14	yes	72	140.107	even	12	inner
140.2.w.b.107.17	yes	72	35.2	odd	12	inner
140.2.w.b.123.14	yes	72	1.1	even	1	trivial
140.2.w.b.123.17	yes	72	4.3	odd	2	inner
700.2.be.e.107.2		72	35.23	odd	12
700.2.be.e.107.5		72	140.23	even	12
700.2.be.e.207.8		72	20.3	even	4
700.2.be.e.207.14		72	5.3	odd	4
700.2.be.e.443.8		72	35.9	even	6
700.2.be.e.443.14		72	140.79	odd	6
700.2.be.e.543.2		72	20.19	odd	2
700.2.be.e.543.5		72	5.4	even	2
980.2.k.j.687.1		36	140.87	odd	12
980.2.k.j.687.8		36	35.17	even	12
980.2.k.j.883.1		36	7.3	odd	6
980.2.k.j.883.8		36	28.3	even	6
980.2.k.k.687.1		36	140.67	even	12
980.2.k.k.687.8		36	35.32	odd	12
980.2.k.k.883.1		36	7.4	even	3
980.2.k.k.883.8		36	28.11	odd	6
980.2.x.m.67.5		72	35.27	even	4
980.2.x.m.67.11		72	140.27	odd	4
980.2.x.m.263.14		72	7.6	odd	2
980.2.x.m.263.17		72	28.27	even	2
980.2.x.m.667.14		72	140.47	odd	12
980.2.x.m.667.17		72	35.12	even	12
980.2.x.m.863.5		72	28.19	even	6
980.2.x.m.863.11		72	7.5	odd	6