Embedded newform 140.2.s.b.59.1

• Label: 140.2.s.b.59.1
• Level: $140$
• Weight: $2$
• Character: 140.59
• 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			59.1
Character	\(\chi\)	\(=\)	140.59
Dual form			140.2.s.b.19.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.41371 - 0.0377920i) q^{2} +(0.634715 + 0.366453i) q^{3} +(1.99714 + 0.106854i) q^{4} +(-0.661137 - 2.13609i) q^{5} +(-0.883452 - 0.542044i) q^{6} +(2.56107 + 0.664037i) q^{7} +(-2.81934 - 0.226536i) q^{8} +(-1.23142 - 2.13289i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 6 q^{4} - 6 q^{5} + 4 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.41371	−	0.0377920i	−0.999643	−	0.0267230i
							\(3\)	0.634715	+	0.366453i	0.366453	+	0.211572i	0.671908	−	0.740635i	\(-0.265475\pi\)
−0.305455	+	0.952207i	\(0.598809\pi\)
\(5\)	−0.661137	−	2.13609i	−0.295670	−	0.955290i
							\(7\)	2.56107	+	0.664037i	0.967992	+	0.250982i
\(11\)	2.33007	+	1.34527i	0.702542	+	0.405613i								0.808294	−	0.588780i	\(-0.200391\pi\)
							−0.105751	+	0.994393i	\(0.533725\pi\)
\(13\)	3.95118			1.09586			0.547930	−	0.836524i	\(-0.315416\pi\)
							0.547930	+	0.836524i	\(0.315416\pi\)
\(17\)	0.709509	−	1.22891i	0.172081	−	0.298053i	−0.767066	−	0.641568i	\(-0.778284\pi\)
							0.939147	+	0.343515i	\(0.111618\pi\)
\(19\)	−1.61265	−	2.79319i	−0.369966	−	0.640801i	0.619593	−	0.784923i	\(-0.287297\pi\)
							−0.989560	+	0.144122i	\(0.953964\pi\)
\(23\)	2.45620	+	4.25426i	0.512152	+	0.887074i	0.999901	+	0.0140897i	\(0.00448504\pi\)
							−0.487748	+	0.872984i	\(0.662182\pi\)
\(29\)	−5.17926			−0.961765			−0.480882	−	0.876785i	\(-0.659684\pi\)
							−0.480882	+	0.876785i	\(0.659684\pi\)
\(31\)	−3.81745	+	6.61201i	−0.685634	+	1.18755i	0.287603	+	0.957750i	\(0.407142\pi\)
							−0.973237	+	0.229803i	\(0.926192\pi\)
\(37\)	−3.87963	+	2.23990i	−0.637806	+	0.368238i	−0.783769	−	0.621052i	\(-0.786705\pi\)
							0.145963	+	0.989290i	\(0.453372\pi\)
\(41\)		−	0.325509i		−	0.0508359i	−0.999677	−	0.0254180i	\(-0.991908\pi\)
							0.999677	−	0.0254180i	\(-0.00809166\pi\)
\(43\)	−9.28165			−1.41544			−0.707719	−	0.706494i	\(-0.750276\pi\)
							−0.707719	+	0.706494i	\(0.750276\pi\)
\(47\)	−5.68610	+	3.28287i	−0.829403	+	0.478856i	−0.853648	−	0.520850i	\(-0.825615\pi\)
							0.0242453	+	0.999706i	\(0.492282\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	140.2.s.b.59.1	yes	32
4.3	odd	2	inner	140.2.s.b.59.6	yes	32
5.2	odd	4		700.2.p.e.451.9		32
5.3	odd	4		700.2.p.e.451.8		32
5.4	even	2	inner	140.2.s.b.59.16	yes	32
7.2	even	3		980.2.s.e.19.11		32
7.3	odd	6		980.2.c.d.979.22		32
7.4	even	3		980.2.c.d.979.21		32
7.5	odd	6	inner	140.2.s.b.19.11	yes	32
7.6	odd	2		980.2.s.e.619.1		32
20.3	even	4		700.2.p.e.451.14		32
20.7	even	4		700.2.p.e.451.3		32
20.19	odd	2	inner	140.2.s.b.59.11	yes	32
28.3	even	6		980.2.c.d.979.9		32
28.11	odd	6		980.2.c.d.979.10		32
28.19	even	6	inner	140.2.s.b.19.16	yes	32
28.23	odd	6		980.2.s.e.19.16		32
28.27	even	2		980.2.s.e.619.6		32
35.4	even	6		980.2.c.d.979.12		32
35.9	even	6		980.2.s.e.19.6		32
35.12	even	12		700.2.p.e.551.3		32
35.19	odd	6	inner	140.2.s.b.19.6	yes	32
35.24	odd	6		980.2.c.d.979.11		32
35.33	even	12		700.2.p.e.551.14		32
35.34	odd	2		980.2.s.e.619.16		32
140.19	even	6	inner	140.2.s.b.19.1	✓	32
140.39	odd	6		980.2.c.d.979.23		32
140.47	odd	12		700.2.p.e.551.9		32
140.59	even	6		980.2.c.d.979.24		32
140.79	odd	6		980.2.s.e.19.1		32
140.103	odd	12		700.2.p.e.551.8		32
140.139	even	2		980.2.s.e.619.11		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
140.2.s.b.19.1	✓	32	140.19	even	6	inner
140.2.s.b.19.6	yes	32	35.19	odd	6	inner
140.2.s.b.19.11	yes	32	7.5	odd	6	inner
140.2.s.b.19.16	yes	32	28.19	even	6	inner
140.2.s.b.59.1	yes	32	1.1	even	1	trivial
140.2.s.b.59.6	yes	32	4.3	odd	2	inner
140.2.s.b.59.11	yes	32	20.19	odd	2	inner
140.2.s.b.59.16	yes	32	5.4	even	2	inner
700.2.p.e.451.3		32	20.7	even	4
700.2.p.e.451.8		32	5.3	odd	4
700.2.p.e.451.9		32	5.2	odd	4
700.2.p.e.451.14		32	20.3	even	4
700.2.p.e.551.3		32	35.12	even	12
700.2.p.e.551.8		32	140.103	odd	12
700.2.p.e.551.9		32	140.47	odd	12
700.2.p.e.551.14		32	35.33	even	12
980.2.c.d.979.9		32	28.3	even	6
980.2.c.d.979.10		32	28.11	odd	6
980.2.c.d.979.11		32	35.24	odd	6
980.2.c.d.979.12		32	35.4	even	6
980.2.c.d.979.21		32	7.4	even	3
980.2.c.d.979.22		32	7.3	odd	6
980.2.c.d.979.23		32	140.39	odd	6
980.2.c.d.979.24		32	140.59	even	6
980.2.s.e.19.1		32	140.79	odd	6
980.2.s.e.19.6		32	35.9	even	6
980.2.s.e.19.11		32	7.2	even	3
980.2.s.e.19.16		32	28.23	odd	6
980.2.s.e.619.1		32	7.6	odd	2
980.2.s.e.619.6		32	28.27	even	2
980.2.s.e.619.11		32	140.139	even	2
980.2.s.e.619.16		32	35.34	odd	2