Newform orbit 70.2.e.d

• Label: 70.2.e.d
• Level: $70$
• Weight: $2$
• Character orbit: 70.e
• Analytic conductor: $0.559$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 70 = 2 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	70.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.558952814149\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
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.

\(f(q)\)	\(=\)	\( q + \zeta_{6} q^{2} + ( - 2 \zeta_{6} + 2) q^{3} + (\zeta_{6} - 1) q^{4} - \zeta_{6} q^{5} + 2 q^{6} + (2 \zeta_{6} - 3) q^{7} - q^{8} - \zeta_{6} q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} + 2 q^{3} - q^{4} - q^{5} + 4 q^{6} - 4 q^{7} - 2 q^{8} - q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/70\mathbb{Z}\right)^\times\).

\(n\)	\(31\)	\(57\)
\(\chi(n)\)	\(-\zeta_{6}\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

11.1

0.500000	+	0.866025i
0.500000	−	0.866025i

0.500000

+

0.866025i

1.00000

−

1.73205i

−0.500000

+

0.866025i

−0.500000

−

0.866025i

2.00000

−2.00000

+

1.73205i

−1.00000

−0.500000

−

0.866025i

0.500000

−

0.866025i

51.1

0.500000

−

0.866025i

1.00000

+

1.73205i

−0.500000

−

0.866025i

−0.500000

+

0.866025i

2.00000

−2.00000

−

1.73205i

−1.00000

−0.500000

+

0.866025i

0.500000

+

0.866025i

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	70.2.e.d	✓	2
3.b	odd	2	1		630.2.k.d		2
4.b	odd	2	1		560.2.q.b		2
5.b	even	2	1		350.2.e.b		2
5.c	odd	4	2		350.2.j.d		4
7.b	odd	2	1		490.2.e.g		2
7.c	even	3	1	inner	70.2.e.d	✓	2
7.c	even	3	1		490.2.a.a		1
7.d	odd	6	1		490.2.a.d		1
7.d	odd	6	1		490.2.e.g		2
21.g	even	6	1		4410.2.a.bg		1
21.h	odd	6	1		630.2.k.d		2
21.h	odd	6	1		4410.2.a.x		1
28.f	even	6	1		3920.2.a.e		1
28.g	odd	6	1		560.2.q.b		2
28.g	odd	6	1		3920.2.a.bh		1
35.i	odd	6	1		2450.2.a.v		1
35.j	even	6	1		350.2.e.b		2
35.j	even	6	1		2450.2.a.bf		1
35.k	even	12	2		2450.2.c.e		2
35.l	odd	12	2		350.2.j.d		4
35.l	odd	12	2		2450.2.c.q		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
70.2.e.d	✓	2	1.a	even	1	1	trivial
70.2.e.d	✓	2	7.c	even	3	1	inner
350.2.e.b		2	5.b	even	2	1
350.2.e.b		2	35.j	even	6	1
350.2.j.d		4	5.c	odd	4	2
350.2.j.d		4	35.l	odd	12	2
490.2.a.a		1	7.c	even	3	1
490.2.a.d		1	7.d	odd	6	1
490.2.e.g		2	7.b	odd	2	1
490.2.e.g		2	7.d	odd	6	1
560.2.q.b		2	4.b	odd	2	1
560.2.q.b		2	28.g	odd	6	1
630.2.k.d		2	3.b	odd	2	1
630.2.k.d		2	21.h	odd	6	1
2450.2.a.v		1	35.i	odd	6	1
2450.2.a.bf		1	35.j	even	6	1
2450.2.c.e		2	35.k	even	12	2
2450.2.c.q		2	35.l	odd	12	2
3920.2.a.e		1	28.f	even	6	1
3920.2.a.bh		1	28.g	odd	6	1
4410.2.a.x		1	21.h	odd	6	1
4410.2.a.bg		1	21.g	even	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(70, [\chi])\):

\( T_{3}^{2} - 2T_{3} + 4 \)

\( T_{13} + 1 \)

Hecke characteristic polynomials

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