Newform orbit 1080.1.p.b

• Label: 1080.1.p.b
• Level: $1080$
• Weight: $1$
• Character orbit: 1080.p
• Analytic conductor: $0.539$
• Analytic rank: $0$
• Dimension: $2$
• Projective image: $D_{6}$
• CM discriminant: -15
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1080 = 2^{3} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1080.p (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.538990213644\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\zeta_{6})\)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{6}\)
Projective field:	Galois closure of 6.2.27993600.1

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\)	\(=\)	q - z^2 * q^2 - z * q^4 + q^5 - q^8 - z^2 * q^10 + z^2 * q^16 + (-z^2 - z) * q^17 + q^19 - z * q^20 - q^23 + q^25 + (z^2 + z) * q^31 + z * q^32 + (-z - 1) * q^34 - z^2 * q^38 - q^40 + z^2 * q^46 - 2 * q^47 - q^49 - z^2 * q^50 + q^53 + (z^2 + z) * q^61 + (z + 1) * q^62 + q^64 + (z^2 - 1) * q^68 - z * q^76 + (-z^2 - z) * q^79 + z^2 * q^80 + (z^2 + z) * q^83 + (-z^2 - z) * q^85 + z * q^92 + 2*z^2 * q^94 + q^95 + z^2 * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q + q^2 - q^4 + 2 * q^5 - 2 * q^8 + q^10 - q^16 + 2 * q^19 - q^20 - 2 * q^23 + 2 * q^25 + q^32 - 3 * q^34 + q^38 - 2 * q^40 - q^46 - 4 * q^47 - 2 * q^49 + q^50 + 2 * q^53 + 3 * q^62 + 2 * q^64 - 3 * q^68 - q^76 - q^80 + q^92 - 2 * q^94 + 2 * q^95 - q^98

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1080\mathbb{Z}\right)^\times\).

\(n\)	\(217\)	\(271\)	\(541\)	\(1001\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(-1\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

379.1

0.500000	+	0.866025i
0.500000	−	0.866025i

0.500000

−

0.866025i

0

−0.500000

−

0.866025i

1.00000

0

0

−1.00000

0

0.500000

−

0.866025i

379.2

0.500000

+

0.866025i

0

−0.500000

+

0.866025i

1.00000

0

0

−1.00000

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 \(\Q(\sqrt{-15}) \)
24.f	even	2	1	inner
40.e	odd	2	1	inner

Twists

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

    

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{23} + 1 \) acting on \(S_{1}^{\mathrm{new}}(1080, [\chi])\).

Hecke characteristic polynomials

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