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

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