Newform orbit 140.2.e.a

• Label: 140.2.e.a
• Level: $140$
• Weight: $2$
• Character orbit: 140.e
• 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.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.11790562830\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-1}) \)
Defining polynomial:	\( x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + 3 i q^{3} + (i - 2) q^{5} - i q^{7} - 6 q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{5} - 12 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(57\)	\(71\)	\(101\)
\(\chi(n)\)	\(-1\)	\(1\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

29.1

	−	1.00000i
		1.00000i

0

−

3.00000i

0

−2.00000

−

1.00000i

0

1.00000i

0

−6.00000

0

29.2

0

3.00000i

0

−2.00000

+

1.00000i

0

−

1.00000i

0

−6.00000

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	140.2.e.a	✓	2
3.b	odd	2	1		1260.2.k.c		2
4.b	odd	2	1		560.2.g.a		2
5.b	even	2	1	inner	140.2.e.a	✓	2
5.c	odd	4	1		700.2.a.a		1
5.c	odd	4	1		700.2.a.j		1
7.b	odd	2	1		980.2.e.b		2
7.c	even	3	2		980.2.q.f		4
7.d	odd	6	2		980.2.q.c		4
8.b	even	2	1		2240.2.g.e		2
8.d	odd	2	1		2240.2.g.f		2
12.b	even	2	1		5040.2.t.s		2
15.d	odd	2	1		1260.2.k.c		2
15.e	even	4	1		6300.2.a.c		1
15.e	even	4	1		6300.2.a.t		1
20.d	odd	2	1		560.2.g.a		2
20.e	even	4	1		2800.2.a.a		1
20.e	even	4	1		2800.2.a.bf		1
35.c	odd	2	1		980.2.e.b		2
35.f	even	4	1		4900.2.a.b		1
35.f	even	4	1		4900.2.a.w		1
35.i	odd	6	2		980.2.q.c		4
35.j	even	6	2		980.2.q.f		4
40.e	odd	2	1		2240.2.g.f		2
40.f	even	2	1		2240.2.g.e		2
60.h	even	2	1		5040.2.t.s		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
140.2.e.a	✓	2	1.a	even	1	1	trivial
140.2.e.a	✓	2	5.b	even	2	1	inner
560.2.g.a		2	4.b	odd	2	1
560.2.g.a		2	20.d	odd	2	1
700.2.a.a		1	5.c	odd	4	1
700.2.a.j		1	5.c	odd	4	1
980.2.e.b		2	7.b	odd	2	1
980.2.e.b		2	35.c	odd	2	1
980.2.q.c		4	7.d	odd	6	2
980.2.q.c		4	35.i	odd	6	2
980.2.q.f		4	7.c	even	3	2
980.2.q.f		4	35.j	even	6	2
1260.2.k.c		2	3.b	odd	2	1
1260.2.k.c		2	15.d	odd	2	1
2240.2.g.e		2	8.b	even	2	1
2240.2.g.e		2	40.f	even	2	1
2240.2.g.f		2	8.d	odd	2	1
2240.2.g.f		2	40.e	odd	2	1
2800.2.a.a		1	20.e	even	4	1
2800.2.a.bf		1	20.e	even	4	1
4900.2.a.b		1	35.f	even	4	1
4900.2.a.w		1	35.f	even	4	1
5040.2.t.s		2	12.b	even	2	1
5040.2.t.s		2	60.h	even	2	1
6300.2.a.c		1	15.e	even	4	1
6300.2.a.t		1	15.e	even	4	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 9 \) acting on \(S_{2}^{\mathrm{new}}(140, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} + 9 \)
$5$	\( T^{2} + 4T + 5 \)
$7$	\( T^{2} + 1 \)
$11$	\( (T - 3)^{2} \)