Newform orbit 70.2.i.a

• Label: 70.2.i.a
• Level: $70$
• Weight: $2$
• Character orbit: 70.i
• Analytic conductor: $0.559$
• Analytic rank: $0$
• Dimension: $4$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 70 = 2 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	70.i (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.558952814149\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Relative dimension:	\(2\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\Q(\zeta_{12})\)
Defining polynomial:	\(x^{4} - x^{2} + 1\)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + \zeta_{12} q^{2} + ( -3 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{3} + \zeta_{12}^{2} q^{4} + ( -1 + 2 \zeta_{12} + \zeta_{12}^{2} ) q^{5} -3 q^{6} + ( \zeta_{12} - 3 \zeta_{12}^{3} ) q^{7} + \zeta_{12}^{3} q^{8} + ( 6 - 6 \zeta_{12}^{2} ) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{4} - 2 q^{5} - 12 q^{6} + 12 q^{9} + O(q^{10}) \)

Character values

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

\(n\)	\(31\)	\(57\)
\(\chi(n)\)	\(-1 + \zeta_{12}^{2}\)	\(-1\)

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

9.1

−0.866025	−	0.500000i
0.866025	+	0.500000i
−0.866025	+	0.500000i
0.866025	−	0.500000i

−0.866025

−

0.500000i

2.59808

−

1.50000i

0.500000

+

0.866025i

−2.23205

−

0.133975i

−3.00000

−0.866025

+

2.50000i

−

1.00000i

3.00000

−

5.19615i

1.86603

+

1.23205i

9.2

0.866025

+

0.500000i

−2.59808

+

1.50000i

0.500000

+

0.866025i

1.23205

+

1.86603i

−3.00000

0.866025

−

2.50000i

1.00000i

3.00000

−

5.19615i

0.133975

+

2.23205i

39.1

−0.866025

+

0.500000i

2.59808

+

1.50000i

0.500000

−

0.866025i

−2.23205

+

0.133975i

−3.00000

−0.866025

−

2.50000i

1.00000i

3.00000

+

5.19615i

1.86603

−

1.23205i

39.2

0.866025

−

0.500000i

−2.59808

−

1.50000i

0.500000

−

0.866025i

1.23205

−

1.86603i

−3.00000

0.866025

+

2.50000i

−

1.00000i

3.00000

+

5.19615i

0.133975

−

2.23205i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
7.c	even	3	1	inner
35.j	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	70.2.i.a	✓	4
3.b	odd	2	1		630.2.u.b		4
4.b	odd	2	1		560.2.bw.c		4
5.b	even	2	1	inner	70.2.i.a	✓	4
5.c	odd	4	1		350.2.e.f		2
5.c	odd	4	1		350.2.e.g		2
7.b	odd	2	1		490.2.i.b		4
7.c	even	3	1	inner	70.2.i.a	✓	4
7.c	even	3	1		490.2.c.c		2
7.d	odd	6	1		490.2.c.b		2
7.d	odd	6	1		490.2.i.b		4
15.d	odd	2	1		630.2.u.b		4
20.d	odd	2	1		560.2.bw.c		4
21.h	odd	6	1		630.2.u.b		4
28.g	odd	6	1		560.2.bw.c		4
35.c	odd	2	1		490.2.i.b		4
35.i	odd	6	1		490.2.c.b		2
35.i	odd	6	1		490.2.i.b		4
35.j	even	6	1	inner	70.2.i.a	✓	4
35.j	even	6	1		490.2.c.c		2
35.k	even	12	1		2450.2.a.c		1
35.k	even	12	1		2450.2.a.bh		1
35.l	odd	12	1		350.2.e.f		2
35.l	odd	12	1		350.2.e.g		2
35.l	odd	12	1		2450.2.a.r		1
35.l	odd	12	1		2450.2.a.s		1
105.o	odd	6	1		630.2.u.b		4
140.p	odd	6	1		560.2.bw.c		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
70.2.i.a	✓	4	1.a	even	1	1	trivial
70.2.i.a	✓	4	5.b	even	2	1	inner
70.2.i.a	✓	4	7.c	even	3	1	inner
70.2.i.a	✓	4	35.j	even	6	1	inner
350.2.e.f		2	5.c	odd	4	1
350.2.e.f		2	35.l	odd	12	1
350.2.e.g		2	5.c	odd	4	1
350.2.e.g		2	35.l	odd	12	1
490.2.c.b		2	7.d	odd	6	1
490.2.c.b		2	35.i	odd	6	1
490.2.c.c		2	7.c	even	3	1
490.2.c.c		2	35.j	even	6	1
490.2.i.b		4	7.b	odd	2	1
490.2.i.b		4	7.d	odd	6	1
490.2.i.b		4	35.c	odd	2	1
490.2.i.b		4	35.i	odd	6	1
560.2.bw.c		4	4.b	odd	2	1
560.2.bw.c		4	20.d	odd	2	1
560.2.bw.c		4	28.g	odd	6	1
560.2.bw.c		4	140.p	odd	6	1
630.2.u.b		4	3.b	odd	2	1
630.2.u.b		4	15.d	odd	2	1
630.2.u.b		4	21.h	odd	6	1
630.2.u.b		4	105.o	odd	6	1
2450.2.a.c		1	35.k	even	12	1
2450.2.a.r		1	35.l	odd	12	1
2450.2.a.s		1	35.l	odd	12	1
2450.2.a.bh		1	35.k	even	12	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 9 T_{3}^{2} + 81 \) acting on \(S_{2}^{\mathrm{new}}(70, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( 1 - T^{2} + T^{4} \)
$3$	\( 81 - 9 T^{2} + T^{4} \)
$5$	\( 25 + 10 T - T^{2} + 2 T^{3} + T^{4} \)
$7$	\( 49 + 11 T^{2} + T^{4} \)
$11$	\( T^{4} \)