Embedded newform 140.2.o.a.131.3

• Label: 140.2.o.a.131.3
• Level: $140$
• Weight: $2$
• Character: 140.131
• 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			131.3
Character	\(\chi\)	\(=\)	140.131
Dual form			140.2.o.a.31.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.26796 + 0.626319i) q^{2} +(-1.49907 - 2.59647i) q^{3} +(1.21545 - 1.58830i) q^{4} +(-0.866025 - 0.500000i) q^{5} +(3.52698 + 2.35332i) q^{6} +(-2.06101 + 1.65899i) q^{7} +(-0.546365 + 2.77516i) q^{8} +(-2.99443 + 5.18651i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 2 q^{2} - 2 q^{4} - 4 q^{8} - 16 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.26796	+	0.626319i	−0.896584	+	0.442874i
							\(3\)	−1.49907	−	2.59647i	−0.865490	−	1.49907i	−0.866560	−	0.499072i	\(-0.833674\pi\)
0.00107081	−	0.999999i	\(-0.499659\pi\)
\(5\)	−0.866025	−	0.500000i	−0.387298	−	0.223607i
							\(7\)	−2.06101	+	1.65899i	−0.778989	+	0.627038i
\(11\)	−1.93693	+	1.11828i	−0.584005	+	0.337175i								−0.762723	−	0.646725i	\(-0.776138\pi\)
							0.178718	+	0.983900i	\(0.442805\pi\)
\(13\)		−	3.17109i		−	0.879502i	−0.898120	−	0.439751i	\(-0.855067\pi\)
							0.898120	−	0.439751i	\(-0.144933\pi\)
\(17\)	−2.98390	+	1.72275i	−0.723701	+	0.417829i	−0.816113	−	0.577892i	\(-0.803876\pi\)
							0.0924124	+	0.995721i	\(0.470542\pi\)
\(19\)	1.02618	−	1.77739i	0.235421	−	0.407761i	−0.723974	−	0.689827i	\(-0.757686\pi\)
							0.959395	+	0.282066i	\(0.0910198\pi\)
\(23\)	−2.30481	−	1.33068i	−0.480587	−	0.277467i	0.240074	−	0.970755i	\(-0.422828\pi\)
							−0.720661	+	0.693288i	\(0.756162\pi\)
\(29\)	−7.38092			−1.37060			−0.685301	−	0.728260i	\(-0.740329\pi\)
							−0.685301	+	0.728260i	\(0.740329\pi\)
\(31\)	−2.44599	−	4.23658i	−0.439313	−	0.760913i	0.558323	−	0.829623i	\(-0.311445\pi\)
							−0.997637	+	0.0687104i	\(0.978112\pi\)
\(37\)	5.59689	−	9.69410i	0.920123	−	1.59370i	0.120902	−	0.992664i	\(-0.461421\pi\)
							0.799222	−	0.601036i	\(-0.205245\pi\)
\(41\)		−	1.46011i		−	0.228031i	−0.993479	−	0.114016i	\(-0.963629\pi\)
							0.993479	−	0.114016i	\(-0.0363714\pi\)
\(43\)			9.95752i			1.51851i	0.650794	+	0.759254i	\(0.274436\pi\)
							−0.650794	+	0.759254i	\(0.725564\pi\)
\(47\)	3.06343	−	5.30601i	0.446847	−	0.773962i	−0.551332	−	0.834286i	\(-0.685880\pi\)
							0.998179	+	0.0603243i	\(0.0192135\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	140.2.o.a.131.3	yes	32
4.3	odd	2	inner	140.2.o.a.131.8	yes	32
5.2	odd	4		700.2.t.d.299.6		32
5.3	odd	4		700.2.t.c.299.11		32
5.4	even	2		700.2.p.c.551.14		32
7.2	even	3		980.2.g.a.391.28		32
7.3	odd	6	inner	140.2.o.a.31.8	yes	32
7.4	even	3		980.2.o.f.31.8		32
7.5	odd	6		980.2.g.a.391.27		32
7.6	odd	2		980.2.o.f.411.3		32
20.3	even	4		700.2.t.c.299.16		32
20.7	even	4		700.2.t.d.299.1		32
20.19	odd	2		700.2.p.c.551.9		32
28.3	even	6	inner	140.2.o.a.31.3	✓	32
28.11	odd	6		980.2.o.f.31.3		32
28.19	even	6		980.2.g.a.391.26		32
28.23	odd	6		980.2.g.a.391.25		32
28.27	even	2		980.2.o.f.411.8		32
35.3	even	12		700.2.t.d.199.1		32
35.17	even	12		700.2.t.c.199.16		32
35.24	odd	6		700.2.p.c.451.9		32
140.3	odd	12		700.2.t.d.199.6		32
140.59	even	6		700.2.p.c.451.14		32
140.87	odd	12		700.2.t.c.199.11		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
140.2.o.a.31.3	✓	32	28.3	even	6	inner
140.2.o.a.31.8	yes	32	7.3	odd	6	inner
140.2.o.a.131.3	yes	32	1.1	even	1	trivial
140.2.o.a.131.8	yes	32	4.3	odd	2	inner
700.2.p.c.451.9		32	35.24	odd	6
700.2.p.c.451.14		32	140.59	even	6
700.2.p.c.551.9		32	20.19	odd	2
700.2.p.c.551.14		32	5.4	even	2
700.2.t.c.199.11		32	140.87	odd	12
700.2.t.c.199.16		32	35.17	even	12
700.2.t.c.299.11		32	5.3	odd	4
700.2.t.c.299.16		32	20.3	even	4
700.2.t.d.199.1		32	35.3	even	12
700.2.t.d.199.6		32	140.3	odd	12
700.2.t.d.299.1		32	20.7	even	4
700.2.t.d.299.6		32	5.2	odd	4
980.2.g.a.391.25		32	28.23	odd	6
980.2.g.a.391.26		32	28.19	even	6
980.2.g.a.391.27		32	7.5	odd	6
980.2.g.a.391.28		32	7.2	even	3
980.2.o.f.31.3		32	28.11	odd	6
980.2.o.f.31.8		32	7.4	even	3
980.2.o.f.411.3		32	7.6	odd	2
980.2.o.f.411.8		32	28.27	even	2