Embedded newform 140.2.s.b.19.5

• Label: 140.2.s.b.19.5
• Level: $140$
• Weight: $2$
• Character: 140.19
• Analytic conductor: $1.118$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 140 = 2^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	140.s (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.11790562830\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(16\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			19.5
Character	\(\chi\)	\(=\)	140.19
Dual form			140.2.s.b.59.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.999687 - 1.00031i) q^{2} +(1.28100 - 0.739583i) q^{3} +(-0.00125109 + 2.00000i) q^{4} +(-1.26649 - 1.84283i) q^{5} +(-2.02041 - 0.542044i) q^{6} +(-0.664037 - 2.56107i) q^{7} +(2.00188 - 1.99812i) q^{8} +(-0.406034 + 0.703271i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 6 q^{4} - 6 q^{5} + 4 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(57\)	\(71\)	\(101\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.999687	−	1.00031i	−0.706886	−	0.707328i
							\(3\)	1.28100	−	0.739583i	0.739583	−	0.426999i	−0.0823346	−	0.996605i	\(-0.526238\pi\)
0.821918	+	0.569606i	\(0.192904\pi\)
\(5\)	−1.26649	−	1.84283i	−0.566390	−	0.824137i
							\(7\)	−0.664037	−	2.56107i	−0.250982	−	0.967992i
\(11\)	5.32875	−	3.07655i	1.60668	−	0.927616i								0.616570	−	0.787300i	\(-0.288522\pi\)
							0.990107	−	0.140316i	\(-0.0448118\pi\)
\(13\)	−3.33366			−0.924591			−0.462295	−	0.886726i	\(-0.652974\pi\)
							−0.462295	+	0.886726i	\(0.652974\pi\)
\(17\)	1.27594	+	2.21000i	0.309462	+	0.536004i	0.978245	−	0.207454i	\(-0.0665178\pi\)
							−0.668783	+	0.743458i	\(0.733184\pi\)
\(19\)	−0.352781	+	0.611035i	−0.0809335	+	0.140181i	−0.903651	−	0.428269i	\(-0.859123\pi\)
							0.822718	+	0.568450i	\(0.192457\pi\)
\(23\)	0.983411	−	1.70332i	0.205055	−	0.355166i	−0.745095	−	0.666958i	\(-0.767596\pi\)
							0.950150	+	0.311792i	\(0.100929\pi\)
\(29\)	5.17926			0.961765			0.480882	−	0.876785i	\(-0.340316\pi\)
							0.480882	+	0.876785i	\(0.340316\pi\)
\(31\)	3.40156	+	5.89167i	0.610937	+	1.05817i	0.991083	+	0.133248i	\(0.0425406\pi\)
							−0.380145	+	0.924927i	\(0.624126\pi\)
\(37\)	5.90431	+	3.40886i	0.970663	+	0.560412i	0.899438	−	0.437048i	\(-0.143976\pi\)
							0.0712245	+	0.997460i	\(0.477309\pi\)
\(41\)		−	2.53373i		−	0.395702i	−0.980232	−	0.197851i	\(-0.936604\pi\)
							0.980232	−	0.197851i	\(-0.0633962\pi\)
\(43\)	4.59892			0.701329			0.350664	−	0.936501i	\(-0.385956\pi\)
							0.350664	+	0.936501i	\(0.385956\pi\)
\(47\)	−3.78388	−	2.18463i	−0.551936	−	0.318660i	0.197966	−	0.980209i	\(-0.436566\pi\)
							−0.749903	+	0.661548i	\(0.769900\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	140.2.s.b.19.5	yes	32
4.3	odd	2	inner	140.2.s.b.19.3	✓	32
5.2	odd	4		700.2.p.e.551.13		32
5.3	odd	4		700.2.p.e.551.4		32
5.4	even	2	inner	140.2.s.b.19.12	yes	32
7.2	even	3		980.2.c.d.979.16		32
7.3	odd	6	inner	140.2.s.b.59.14	yes	32
7.4	even	3		980.2.s.e.619.14		32
7.5	odd	6		980.2.c.d.979.15		32
7.6	odd	2		980.2.s.e.19.5		32
20.3	even	4		700.2.p.e.551.6		32
20.7	even	4		700.2.p.e.551.11		32
20.19	odd	2	inner	140.2.s.b.19.14	yes	32
28.3	even	6	inner	140.2.s.b.59.12	yes	32
28.11	odd	6		980.2.s.e.619.12		32
28.19	even	6		980.2.c.d.979.20		32
28.23	odd	6		980.2.c.d.979.19		32
28.27	even	2		980.2.s.e.19.3		32
35.3	even	12		700.2.p.e.451.6		32
35.4	even	6		980.2.s.e.619.3		32
35.9	even	6		980.2.c.d.979.17		32
35.17	even	12		700.2.p.e.451.11		32
35.19	odd	6		980.2.c.d.979.18		32
35.24	odd	6	inner	140.2.s.b.59.3	yes	32
35.34	odd	2		980.2.s.e.19.12		32
140.3	odd	12		700.2.p.e.451.4		32
140.19	even	6		980.2.c.d.979.13		32
140.39	odd	6		980.2.s.e.619.5		32
140.59	even	6	inner	140.2.s.b.59.5	yes	32
140.79	odd	6		980.2.c.d.979.14		32
140.87	odd	12		700.2.p.e.451.13		32
140.139	even	2		980.2.s.e.19.14		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
140.2.s.b.19.3	✓	32	4.3	odd	2	inner
140.2.s.b.19.5	yes	32	1.1	even	1	trivial
140.2.s.b.19.12	yes	32	5.4	even	2	inner
140.2.s.b.19.14	yes	32	20.19	odd	2	inner
140.2.s.b.59.3	yes	32	35.24	odd	6	inner
140.2.s.b.59.5	yes	32	140.59	even	6	inner
140.2.s.b.59.12	yes	32	28.3	even	6	inner
140.2.s.b.59.14	yes	32	7.3	odd	6	inner
700.2.p.e.451.4		32	140.3	odd	12
700.2.p.e.451.6		32	35.3	even	12
700.2.p.e.451.11		32	35.17	even	12
700.2.p.e.451.13		32	140.87	odd	12
700.2.p.e.551.4		32	5.3	odd	4
700.2.p.e.551.6		32	20.3	even	4
700.2.p.e.551.11		32	20.7	even	4
700.2.p.e.551.13		32	5.2	odd	4
980.2.c.d.979.13		32	140.19	even	6
980.2.c.d.979.14		32	140.79	odd	6
980.2.c.d.979.15		32	7.5	odd	6
980.2.c.d.979.16		32	7.2	even	3
980.2.c.d.979.17		32	35.9	even	6
980.2.c.d.979.18		32	35.19	odd	6
980.2.c.d.979.19		32	28.23	odd	6
980.2.c.d.979.20		32	28.19	even	6
980.2.s.e.19.3		32	28.27	even	2
980.2.s.e.19.5		32	7.6	odd	2
980.2.s.e.19.12		32	35.34	odd	2
980.2.s.e.19.14		32	140.139	even	2
980.2.s.e.619.3		32	35.4	even	6
980.2.s.e.619.5		32	140.39	odd	6
980.2.s.e.619.12		32	28.11	odd	6
980.2.s.e.619.14		32	7.4	even	3