Embedded newform 140.2.o.a.31.15

• Label: 140.2.o.a.31.15
• Level: $140$
• Weight: $2$
• Character: 140.31
• Analytic conductor: $1.118$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 140 = 2^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	140.o (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			31.15
Character	\(\chi\)	\(=\)	140.31
Dual form			140.2.o.a.131.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.37405 + 0.334637i) q^{2} +(-0.556469 + 0.963833i) q^{3} +(1.77604 + 0.919616i) q^{4} +(0.866025 - 0.500000i) q^{5} +(-1.08715 + 1.13814i) q^{6} +(-2.32410 - 1.26433i) q^{7} +(2.13263 + 1.85793i) q^{8} +(0.880685 + 1.52539i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 2 q^{2} - 2 q^{4} - 4 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.37405	+	0.334637i	0.971601	+	0.236624i
							\(3\)	−0.556469	+	0.963833i	−0.321278	+	0.556469i	−0.980752	−	0.195258i	\(-0.937446\pi\)
0.659474	+	0.751727i	\(0.270779\pi\)
\(5\)	0.866025	−	0.500000i	0.387298	−	0.223607i
							\(7\)	−2.32410	−	1.26433i	−0.878429	−	0.477874i
\(11\)	−1.48417	−	0.856885i	−0.447493	−	0.258360i								0.259278	−	0.965803i	\(-0.416516\pi\)
							−0.706771	+	0.707442i	\(0.749849\pi\)
\(13\)		−	2.45950i		−	0.682142i	−0.940038	−	0.341071i	\(-0.889210\pi\)
							0.940038	−	0.341071i	\(-0.110790\pi\)
\(17\)	−5.38574	−	3.10946i	−1.30623	−	0.754155i	−0.324768	−	0.945794i	\(-0.605286\pi\)
							−0.981465	+	0.191639i	\(0.938620\pi\)
\(19\)	0.108256	+	0.187505i	0.0248357	+	0.0430167i	0.878176	−	0.478337i	\(-0.158760\pi\)
							−0.853340	+	0.521354i	\(0.825427\pi\)
\(23\)	5.68681	−	3.28328i	1.18578	−	0.684612i	0.228437	−	0.973559i	\(-0.426638\pi\)
							0.957345	+	0.288947i	\(0.0933051\pi\)
\(29\)	−2.47123			−0.458896			−0.229448	−	0.973321i	\(-0.573692\pi\)
							−0.229448	+	0.973321i	\(0.573692\pi\)
\(31\)	−0.0819131	+	0.141878i	−0.0147120	+	0.0254820i	−0.873288	−	0.487205i	\(-0.838016\pi\)
							0.858576	+	0.512687i	\(0.171350\pi\)
\(37\)	−3.84840	−	6.66562i	−0.632673	−	1.09582i	−0.987003	−	0.160701i	\(-0.948625\pi\)
							0.354331	−	0.935120i	\(-0.384709\pi\)
\(41\)			8.34130i			1.30269i	0.758781	+	0.651346i	\(0.225795\pi\)
							−0.758781	+	0.651346i	\(0.774205\pi\)
\(43\)			1.89449i			0.288907i	0.989512	+	0.144454i	\(0.0461425\pi\)
							−0.989512	+	0.144454i	\(0.953858\pi\)
\(47\)	5.85225	+	10.1364i	0.853639	+	1.47855i	0.877902	+	0.478840i	\(0.158943\pi\)
							−0.0242630	+	0.999706i	\(0.507724\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	140.2.o.a.31.15	yes	32
4.3	odd	2	inner	140.2.o.a.31.7	✓	32
5.2	odd	4		700.2.t.d.199.7		32
5.3	odd	4		700.2.t.c.199.10		32
5.4	even	2		700.2.p.c.451.2		32
7.2	even	3		980.2.o.f.411.7		32
7.3	odd	6		980.2.g.a.391.11		32
7.4	even	3		980.2.g.a.391.12		32
7.5	odd	6	inner	140.2.o.a.131.7	yes	32
7.6	odd	2		980.2.o.f.31.15		32
20.3	even	4		700.2.t.c.199.15		32
20.7	even	4		700.2.t.d.199.2		32
20.19	odd	2		700.2.p.c.451.10		32
28.3	even	6		980.2.g.a.391.10		32
28.11	odd	6		980.2.g.a.391.9		32
28.19	even	6	inner	140.2.o.a.131.15	yes	32
28.23	odd	6		980.2.o.f.411.15		32
28.27	even	2		980.2.o.f.31.7		32
35.12	even	12		700.2.t.c.299.15		32
35.19	odd	6		700.2.p.c.551.10		32
35.33	even	12		700.2.t.d.299.2		32
140.19	even	6		700.2.p.c.551.2		32
140.47	odd	12		700.2.t.c.299.10		32
140.103	odd	12		700.2.t.d.299.7		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
140.2.o.a.31.7	✓	32	4.3	odd	2	inner
140.2.o.a.31.15	yes	32	1.1	even	1	trivial
140.2.o.a.131.7	yes	32	7.5	odd	6	inner
140.2.o.a.131.15	yes	32	28.19	even	6	inner
700.2.p.c.451.2		32	5.4	even	2
700.2.p.c.451.10		32	20.19	odd	2
700.2.p.c.551.2		32	140.19	even	6
700.2.p.c.551.10		32	35.19	odd	6
700.2.t.c.199.10		32	5.3	odd	4
700.2.t.c.199.15		32	20.3	even	4
700.2.t.c.299.10		32	140.47	odd	12
700.2.t.c.299.15		32	35.12	even	12
700.2.t.d.199.2		32	20.7	even	4
700.2.t.d.199.7		32	5.2	odd	4
700.2.t.d.299.2		32	35.33	even	12
700.2.t.d.299.7		32	140.103	odd	12
980.2.g.a.391.9		32	28.11	odd	6
980.2.g.a.391.10		32	28.3	even	6
980.2.g.a.391.11		32	7.3	odd	6
980.2.g.a.391.12		32	7.4	even	3
980.2.o.f.31.7		32	28.27	even	2
980.2.o.f.31.15		32	7.6	odd	2
980.2.o.f.411.7		32	7.2	even	3
980.2.o.f.411.15		32	28.23	odd	6