Embedded newform 140.2.s.b.19.15

• Label: 140.2.s.b.19.15
• 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			19.15
Character	\(\chi\)	\(=\)	140.19
Dual form			140.2.s.b.59.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.38503 - 0.285823i) q^{2} +(-2.24836 + 1.29809i) q^{3} +(1.83661 - 0.791746i) q^{4} +(0.316513 + 2.21355i) q^{5} +(-2.74302 + 2.44053i) q^{6} +(2.57589 + 0.603960i) q^{7} +(2.31746 - 1.62154i) q^{8} +(1.87009 - 3.23909i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 6 q^{4} - 6 q^{5} + 4 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.38503	−	0.285823i	0.979363	−	0.202107i
							\(3\)	−2.24836	+	1.29809i	−1.29809	+	0.749454i	−0.980075	−	0.198630i	\(-0.936351\pi\)
−0.318019	+	0.948084i	\(0.603017\pi\)
\(5\)	0.316513	+	2.21355i	0.141549	+	0.989931i
							\(7\)	2.57589	+	0.603960i	0.973597	+	0.228275i
\(11\)	−3.12008	+	1.80138i	−0.940738	+	0.543136i								−0.890192	−	0.455586i	\(-0.849430\pi\)
							−0.0505467	+	0.998722i	\(0.516096\pi\)
\(13\)	0.818282			0.226951			0.113475	−	0.993541i	\(-0.463802\pi\)
							0.113475	+	0.993541i	\(0.463802\pi\)
\(17\)	−3.69814	−	6.40537i	−0.896931	−	1.55353i	−0.831396	−	0.555680i	\(-0.812458\pi\)
							−0.0655347	−	0.997850i	\(-0.520875\pi\)
\(19\)	1.65329	−	2.86358i	0.379291	−	0.656951i	−0.611668	−	0.791114i	\(-0.709501\pi\)
							0.990959	+	0.134163i	\(0.0428346\pi\)
\(23\)	1.26758	−	2.19550i	0.264308	−	0.457794i	−0.703074	−	0.711116i	\(-0.748190\pi\)
							0.967382	+	0.253322i	\(0.0815232\pi\)
\(29\)	−2.04334			−0.379439			−0.189720	−	0.981838i	\(-0.560758\pi\)
							−0.189720	+	0.981838i	\(0.560758\pi\)
\(31\)	0.955727	+	1.65537i	0.171654	+	0.297313i	0.938998	−	0.343922i	\(-0.111756\pi\)
							−0.767345	+	0.641235i	\(0.778422\pi\)
\(37\)	6.20697	+	3.58360i	1.02042	+	0.589140i	0.914226	−	0.405206i	\(-0.132800\pi\)
							0.106195	+	0.994345i	\(0.466133\pi\)
\(41\)			2.65824i			0.415147i	0.978219	+	0.207573i	\(0.0665566\pi\)
							−0.978219	+	0.207573i	\(0.933443\pi\)
\(43\)	−2.39696			−0.365533			−0.182766	−	0.983156i	\(-0.558505\pi\)
							−0.182766	+	0.983156i	\(0.558505\pi\)
\(47\)	1.15593	+	0.667376i	0.168610	+	0.0973468i	0.581930	−	0.813239i	\(-0.302298\pi\)
							−0.413320	+	0.910586i	\(0.635631\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	140.2.s.b.19.15	yes	32
4.3	odd	2	inner	140.2.s.b.19.10	yes	32
5.2	odd	4		700.2.p.e.551.10		32
5.3	odd	4		700.2.p.e.551.7		32
5.4	even	2	inner	140.2.s.b.19.2	✓	32
7.2	even	3		980.2.c.d.979.5		32
7.3	odd	6	inner	140.2.s.b.59.7	yes	32
7.4	even	3		980.2.s.e.619.7		32
7.5	odd	6		980.2.c.d.979.6		32
7.6	odd	2		980.2.s.e.19.15		32
20.3	even	4		700.2.p.e.551.15		32
20.7	even	4		700.2.p.e.551.2		32
20.19	odd	2	inner	140.2.s.b.19.7	yes	32
28.3	even	6	inner	140.2.s.b.59.2	yes	32
28.11	odd	6		980.2.s.e.619.2		32
28.19	even	6		980.2.c.d.979.25		32
28.23	odd	6		980.2.c.d.979.26		32
28.27	even	2		980.2.s.e.19.10		32
35.3	even	12		700.2.p.e.451.15		32
35.4	even	6		980.2.s.e.619.10		32
35.9	even	6		980.2.c.d.979.28		32
35.17	even	12		700.2.p.e.451.2		32
35.19	odd	6		980.2.c.d.979.27		32
35.24	odd	6	inner	140.2.s.b.59.10	yes	32
35.34	odd	2		980.2.s.e.19.2		32
140.3	odd	12		700.2.p.e.451.7		32
140.19	even	6		980.2.c.d.979.8		32
140.39	odd	6		980.2.s.e.619.15		32
140.59	even	6	inner	140.2.s.b.59.15	yes	32
140.79	odd	6		980.2.c.d.979.7		32
140.87	odd	12		700.2.p.e.451.10		32
140.139	even	2		980.2.s.e.19.7		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
140.2.s.b.19.2	✓	32	5.4	even	2	inner
140.2.s.b.19.7	yes	32	20.19	odd	2	inner
140.2.s.b.19.10	yes	32	4.3	odd	2	inner
140.2.s.b.19.15	yes	32	1.1	even	1	trivial
140.2.s.b.59.2	yes	32	28.3	even	6	inner
140.2.s.b.59.7	yes	32	7.3	odd	6	inner
140.2.s.b.59.10	yes	32	35.24	odd	6	inner
140.2.s.b.59.15	yes	32	140.59	even	6	inner
700.2.p.e.451.2		32	35.17	even	12
700.2.p.e.451.7		32	140.3	odd	12
700.2.p.e.451.10		32	140.87	odd	12
700.2.p.e.451.15		32	35.3	even	12
700.2.p.e.551.2		32	20.7	even	4
700.2.p.e.551.7		32	5.3	odd	4
700.2.p.e.551.10		32	5.2	odd	4
700.2.p.e.551.15		32	20.3	even	4
980.2.c.d.979.5		32	7.2	even	3
980.2.c.d.979.6		32	7.5	odd	6
980.2.c.d.979.7		32	140.79	odd	6
980.2.c.d.979.8		32	140.19	even	6
980.2.c.d.979.25		32	28.19	even	6
980.2.c.d.979.26		32	28.23	odd	6
980.2.c.d.979.27		32	35.19	odd	6
980.2.c.d.979.28		32	35.9	even	6
980.2.s.e.19.2		32	35.34	odd	2
980.2.s.e.19.7		32	140.139	even	2
980.2.s.e.19.10		32	28.27	even	2
980.2.s.e.19.15		32	7.6	odd	2
980.2.s.e.619.2		32	28.11	odd	6
980.2.s.e.619.7		32	7.4	even	3
980.2.s.e.619.10		32	35.4	even	6
980.2.s.e.619.15		32	140.39	odd	6