Newform orbit 3240.1.bh.g

• Label: 3240.1.bh.g
• Level: $3240$
• Weight: $1$
• Character orbit: 3240.bh
• Analytic conductor: $1.617$
• Analytic rank: $0$
• Dimension: $4$
• Projective image: $D_{6}$
• CM discriminant: -15
• Inner twists: $8$

Newspace parameters

Level:	$N$	$=$	$3240 = 2^{3} \cdot 3^{4} \cdot 5$
Weight:	$k$	$=$	$1$
Character orbit:	$[\chi]$	$=$	3240.bh (of order $6$, degree $2$, not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$1.61697064093$
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:	no (minimal twist has level 1080)
Projective image:	$D_{6}$
Projective field:	Galois closure of 6.0.27993600.2

$q$-expansion

The $q$-expansion and trace form are shown below.

$f(q)$	$=$	q + z^5 * q^2 - z^4 * q^4 - z^5 * q^5 + z^3 * q^8 + z^4 * q^10 - z^2 * q^16 + (z^5 - z) * q^17 + (-z^4 - z^2) * q^19 - z^3 * q^20 + (-z^3 - z) * q^23 - z^4 * q^25 - z^2 * q^31 + z * q^32 + (-z^4 + 1) * q^34 + (z^3 + z) * q^38 + z^2 * q^40 + (z^2 + 1) * q^46 - z^2 * q^49 + z^3 * q^50 + z^3 * q^53 + (z^2 + 1) * q^61 + z * q^62 - q^64 + (z^5 + z^3) * q^68 + (-z^2 - 1) * q^76 - z^4 * q^79 - z * q^80 + z * q^83 + (z^4 - 1) * q^85 + (z^5 - z) * q^92 + (-z^3 - z) * q^95 + z * q^98
$\operatorname{Tr}(f)(q)$	$=$	$4 q + 2 q^{4} - 2 q^{10} - 2 q^{16} + 2 q^{25} - 2 q^{31} + 6 q^{34} + 2 q^{40} + 6 q^{46} - 2 q^{49} + 6 q^{61} - 4 q^{64} - 6 q^{76} + 2 q^{79} - 6 q^{85}+O(q^{100})$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3240\mathbb{Z}\right)^\times$.

$n$	$1297$	$1621$	$2431$	$3161$
$\chi(n)$	$-1$	$-1$	$1$	$\zeta_{12}^{2}$

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  

$\iota_m(\nu)$

$a_{2}$

$a_{3}$

$a_{4}$

$a_{5}$

$a_{6}$

$a_{7}$

$a_{8}$

$a_{9}$

$a_{10}$

269.1

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

−0.866025

+

0.500000i

0

0.500000

−

0.866025i

0.866025

−

0.500000i

0

0

1.00000i

0

−0.500000

+

0.866025i

269.2

0.866025

−

0.500000i

0

0.500000

−

0.866025i

−0.866025

+

0.500000i

0

0

−

1.00000i

0

−0.500000

+

0.866025i

1349.1

−0.866025

−

0.500000i

0

0.500000

+

0.866025i

0.866025

+

0.500000i

0

0

−

1.00000i

0

−0.500000

−

0.866025i

1349.2

0.866025

+

0.500000i

0

0.500000

+

0.866025i

−0.866025

−

0.500000i

0

0

1.00000i

0

−0.500000

−

0.866025i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
15.d	odd	2	1	CM by $\Q(\sqrt{-15})$
3.b	odd	2	1	inner
5.b	even	2	1	inner
72.j	odd	6	1	inner
72.n	even	6	1	inner
360.bh	odd	6	1	inner
360.bk	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3240.1.bh.g		4
3.b	odd	2	1	inner	3240.1.bh.g		4
5.b	even	2	1	inner	3240.1.bh.g		4
8.b	even	2	1		3240.1.bh.e		4
9.c	even	3	1		1080.1.i.f	✓	4
9.c	even	3	1		3240.1.bh.e		4
9.d	odd	6	1		1080.1.i.f	✓	4
9.d	odd	6	1		3240.1.bh.e		4
15.d	odd	2	1	CM	3240.1.bh.g		4
24.h	odd	2	1		3240.1.bh.e		4
40.f	even	2	1		3240.1.bh.e		4
45.h	odd	6	1		1080.1.i.f	✓	4
45.h	odd	6	1		3240.1.bh.e		4
45.j	even	6	1		1080.1.i.f	✓	4
45.j	even	6	1		3240.1.bh.e		4
72.j	odd	6	1		1080.1.i.f	✓	4
72.j	odd	6	1	inner	3240.1.bh.g		4
72.n	even	6	1		1080.1.i.f	✓	4
72.n	even	6	1	inner	3240.1.bh.g		4
120.i	odd	2	1		3240.1.bh.e		4
360.bh	odd	6	1		1080.1.i.f	✓	4
360.bh	odd	6	1	inner	3240.1.bh.g		4
360.bk	even	6	1		1080.1.i.f	✓	4
360.bk	even	6	1	inner	3240.1.bh.g		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1080.1.i.f	✓	4	9.c	even	3	1
1080.1.i.f	✓	4	9.d	odd	6	1
1080.1.i.f	✓	4	45.h	odd	6	1
1080.1.i.f	✓	4	45.j	even	6	1
1080.1.i.f	✓	4	72.j	odd	6	1
1080.1.i.f	✓	4	72.n	even	6	1
1080.1.i.f	✓	4	360.bh	odd	6	1
1080.1.i.f	✓	4	360.bk	even	6	1
3240.1.bh.e		4	8.b	even	2	1
3240.1.bh.e		4	9.c	even	3	1
3240.1.bh.e		4	9.d	odd	6	1
3240.1.bh.e		4	24.h	odd	2	1
3240.1.bh.e		4	40.f	even	2	1
3240.1.bh.e		4	45.h	odd	6	1
3240.1.bh.e		4	45.j	even	6	1
3240.1.bh.e		4	120.i	odd	2	1
3240.1.bh.g		4	1.a	even	1	1	trivial
3240.1.bh.g		4	3.b	odd	2	1	inner
3240.1.bh.g		4	5.b	even	2	1	inner
3240.1.bh.g		4	15.d	odd	2	1	CM
3240.1.bh.g		4	72.j	odd	6	1	inner
3240.1.bh.g		4	72.n	even	6	1	inner
3240.1.bh.g		4	360.bh	odd	6	1	inner
3240.1.bh.g		4	360.bk	even	6	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{1}^{\mathrm{new}}(3240, [\chi])$:

$T_{7}$

$T_{11}$

$T_{13}$

$T_{17}^{2} - 3$

$T_{61}^{2} - 3T_{61} + 3$

Hecke characteristic polynomials

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