Newform orbit 140.2.i.a

• Label: 140.2.i.a
• Level: $140$
• Weight: $2$
• Character orbit: 140.i
• Analytic conductor: $1.118$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 140 = 2^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	140.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.11790562830\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + (\zeta_{6} - 1) q^{3} + \zeta_{6} q^{5} + ( - \zeta_{6} + 3) q^{7} + 2 \zeta_{6} q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} + q^{5} + 5 q^{7} + 2 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(57\)	\(71\)	\(101\)
\(\chi(n)\)	\(1\)	\(1\)	\(-\zeta_{6}\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

81.1

0.500000	+	0.866025i
0.500000	−	0.866025i

0

−0.500000

+

0.866025i

0

0.500000

+

0.866025i

0

2.50000

−

0.866025i

0

1.00000

+

1.73205i

0

121.1

0

−0.500000

−

0.866025i

0

0.500000

−

0.866025i

0

2.50000

+

0.866025i

0

1.00000

−

1.73205i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	140.2.i.a	✓	2
3.b	odd	2	1		1260.2.s.c		2
4.b	odd	2	1		560.2.q.f		2
5.b	even	2	1		700.2.i.b		2
5.c	odd	4	2		700.2.r.a		4
7.b	odd	2	1		980.2.i.f		2
7.c	even	3	1	inner	140.2.i.a	✓	2
7.c	even	3	1		980.2.a.g		1
7.d	odd	6	1		980.2.a.e		1
7.d	odd	6	1		980.2.i.f		2
21.g	even	6	1		8820.2.a.a		1
21.h	odd	6	1		1260.2.s.c		2
21.h	odd	6	1		8820.2.a.p		1
28.f	even	6	1		3920.2.a.w		1
28.g	odd	6	1		560.2.q.f		2
28.g	odd	6	1		3920.2.a.k		1
35.i	odd	6	1		4900.2.a.q		1
35.j	even	6	1		700.2.i.b		2
35.j	even	6	1		4900.2.a.i		1
35.k	even	12	2		4900.2.e.n		2
35.l	odd	12	2		700.2.r.a		4
35.l	odd	12	2		4900.2.e.m		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
140.2.i.a	✓	2	1.a	even	1	1	trivial
140.2.i.a	✓	2	7.c	even	3	1	inner
560.2.q.f		2	4.b	odd	2	1
560.2.q.f		2	28.g	odd	6	1
700.2.i.b		2	5.b	even	2	1
700.2.i.b		2	35.j	even	6	1
700.2.r.a		4	5.c	odd	4	2
700.2.r.a		4	35.l	odd	12	2
980.2.a.e		1	7.d	odd	6	1
980.2.a.g		1	7.c	even	3	1
980.2.i.f		2	7.b	odd	2	1
980.2.i.f		2	7.d	odd	6	1
1260.2.s.c		2	3.b	odd	2	1
1260.2.s.c		2	21.h	odd	6	1
3920.2.a.k		1	28.g	odd	6	1
3920.2.a.w		1	28.f	even	6	1
4900.2.a.i		1	35.j	even	6	1
4900.2.a.q		1	35.i	odd	6	1
4900.2.e.m		2	35.l	odd	12	2
4900.2.e.n		2	35.k	even	12	2
8820.2.a.a		1	21.g	even	6	1
8820.2.a.p		1	21.h	odd	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + T_{3} + 1 \) acting on \(S_{2}^{\mathrm{new}}(140, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} + T + 1 \)
$5$	\( T^{2} - T + 1 \)
$7$	\( T^{2} - 5T + 7 \)
$11$	\( T^{2} + 6T + 36 \)