Embedded newform 140.2.s.b.19.9

• Label: 140.2.s.b.19.9
• 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			19.9
Character	\(\chi\)	\(=\)	140.19
Dual form			140.2.s.b.59.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.163027 - 1.40479i) q^{2} +(1.62820 - 0.940044i) q^{3} +(-1.94684 - 0.458035i) q^{4} +(0.430625 - 2.19421i) q^{5} +(-1.05512 - 2.44053i) q^{6} +(-0.603960 + 2.57589i) q^{7} +(-0.960828 + 2.66023i) q^{8} +(0.267367 - 0.463092i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 6 q^{4} - 6 q^{5} + 4 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.163027	−	1.40479i	0.115277	−	0.993333i
							\(3\)	1.62820	−	0.940044i	0.940044	−	0.542735i	0.0500701	−	0.998746i	\(-0.484056\pi\)
0.889974	+	0.456011i	\(0.150722\pi\)
\(5\)	0.430625	−	2.19421i	0.192581	−	0.981281i
							\(7\)	−0.603960	+	2.57589i	−0.228275	+	0.973597i
\(11\)	−1.20016	+	0.692910i	−0.361860	+	0.208920i								−0.669897	−	0.742454i	\(-0.733662\pi\)
							0.308036	+	0.951375i	\(0.400328\pi\)
\(13\)	5.79701			1.60780			0.803901	−	0.594764i	\(-0.202754\pi\)
							0.803901	+	0.594764i	\(0.202754\pi\)
\(17\)	−2.33288	−	4.04066i	−0.565806	−	0.980004i	−0.996974	−	0.0777328i	\(-0.975232\pi\)
							0.431169	−	0.902271i	\(-0.358101\pi\)
\(19\)	−2.74619	+	4.75654i	−0.630019	+	1.09122i	0.357529	+	0.933902i	\(0.383619\pi\)
							−0.987547	+	0.157322i	\(0.949714\pi\)
\(23\)	2.32234	−	4.02241i	0.484241	−	0.838730i	−0.515595	−	0.856832i	\(-0.672429\pi\)
							0.999836	+	0.0181024i	\(0.00576250\pi\)
\(29\)	2.04334			0.379439			0.189720	−	0.981838i	\(-0.439242\pi\)
							0.189720	+	0.981838i	\(0.439242\pi\)
\(31\)	0.832529	+	1.44198i	0.149527	+	0.258988i	0.931053	−	0.364885i	\(-0.118892\pi\)
							−0.781526	+	0.623873i	\(0.785558\pi\)
\(37\)	0.247194	+	0.142718i	0.0406385	+	0.0234627i	0.520182	−	0.854056i	\(-0.325864\pi\)
							−0.479543	+	0.877518i	\(0.659198\pi\)
\(41\)			8.21024i			1.28222i	0.767447	+	0.641112i	\(0.221527\pi\)
							−0.767447	+	0.641112i	\(0.778473\pi\)
\(43\)	−3.30992			−0.504758			−0.252379	−	0.967628i	\(-0.581213\pi\)
							−0.252379	+	0.967628i	\(0.581213\pi\)
\(47\)	−0.119679	−	0.0690967i	−0.0174570	−	0.0100788i	0.491246	−	0.871021i	\(-0.336542\pi\)
							−0.508703	+	0.860942i	\(0.669875\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	140.2.s.b.19.9	yes	32
4.3	odd	2	inner	140.2.s.b.19.4	✓	32
5.2	odd	4		700.2.p.e.551.16		32
5.3	odd	4		700.2.p.e.551.1		32
5.4	even	2	inner	140.2.s.b.19.8	yes	32
7.2	even	3		980.2.c.d.979.4		32
7.3	odd	6	inner	140.2.s.b.59.13	yes	32
7.4	even	3		980.2.s.e.619.13		32
7.5	odd	6		980.2.c.d.979.3		32
7.6	odd	2		980.2.s.e.19.9		32
20.3	even	4		700.2.p.e.551.12		32
20.7	even	4		700.2.p.e.551.5		32
20.19	odd	2	inner	140.2.s.b.19.13	yes	32
28.3	even	6	inner	140.2.s.b.59.8	yes	32
28.11	odd	6		980.2.s.e.619.8		32
28.19	even	6		980.2.c.d.979.32		32
28.23	odd	6		980.2.c.d.979.31		32
28.27	even	2		980.2.s.e.19.4		32
35.3	even	12		700.2.p.e.451.12		32
35.4	even	6		980.2.s.e.619.4		32
35.9	even	6		980.2.c.d.979.29		32
35.17	even	12		700.2.p.e.451.5		32
35.19	odd	6		980.2.c.d.979.30		32
35.24	odd	6	inner	140.2.s.b.59.4	yes	32
35.34	odd	2		980.2.s.e.19.8		32
140.3	odd	12		700.2.p.e.451.1		32
140.19	even	6		980.2.c.d.979.1		32
140.39	odd	6		980.2.s.e.619.9		32
140.59	even	6	inner	140.2.s.b.59.9	yes	32
140.79	odd	6		980.2.c.d.979.2		32
140.87	odd	12		700.2.p.e.451.16		32
140.139	even	2		980.2.s.e.19.13		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
140.2.s.b.19.4	✓	32	4.3	odd	2	inner
140.2.s.b.19.8	yes	32	5.4	even	2	inner
140.2.s.b.19.9	yes	32	1.1	even	1	trivial
140.2.s.b.19.13	yes	32	20.19	odd	2	inner
140.2.s.b.59.4	yes	32	35.24	odd	6	inner
140.2.s.b.59.8	yes	32	28.3	even	6	inner
140.2.s.b.59.9	yes	32	140.59	even	6	inner
140.2.s.b.59.13	yes	32	7.3	odd	6	inner
700.2.p.e.451.1		32	140.3	odd	12
700.2.p.e.451.5		32	35.17	even	12
700.2.p.e.451.12		32	35.3	even	12
700.2.p.e.451.16		32	140.87	odd	12
700.2.p.e.551.1		32	5.3	odd	4
700.2.p.e.551.5		32	20.7	even	4
700.2.p.e.551.12		32	20.3	even	4
700.2.p.e.551.16		32	5.2	odd	4
980.2.c.d.979.1		32	140.19	even	6
980.2.c.d.979.2		32	140.79	odd	6
980.2.c.d.979.3		32	7.5	odd	6
980.2.c.d.979.4		32	7.2	even	3
980.2.c.d.979.29		32	35.9	even	6
980.2.c.d.979.30		32	35.19	odd	6
980.2.c.d.979.31		32	28.23	odd	6
980.2.c.d.979.32		32	28.19	even	6
980.2.s.e.19.4		32	28.27	even	2
980.2.s.e.19.8		32	35.34	odd	2
980.2.s.e.19.9		32	7.6	odd	2
980.2.s.e.19.13		32	140.139	even	2
980.2.s.e.619.4		32	35.4	even	6
980.2.s.e.619.8		32	28.11	odd	6
980.2.s.e.619.9		32	140.39	odd	6
980.2.s.e.619.13		32	7.4	even	3