Newform orbit 2340.1.db.e

• Label: 2340.1.db.e
• Level: $2340$
• Weight: $1$
• Character orbit: 2340.db
• Analytic conductor: $1.168$
• Analytic rank: $0$
• Dimension: $4$
• Projective image: $D_{6}$
• CM discriminant: -260
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 2340 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2340.db (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.16781212956\)
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:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{6}\)
Projective field:	Galois closure of 6.0.1774094400.1

$q$-expansion

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

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2340\mathbb{Z}\right)^\times\).

\(n\)	\(937\)	\(1081\)	\(1171\)	\(2081\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(-1\)	\(-\zeta_{12}^{2}\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

259.1

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

−0.500000

+

0.866025i

−

1.00000i

−0.500000

−

0.866025i

0.500000

+

0.866025i

0.866025

+

0.500000i

0

1.00000

−1.00000

−1.00000

259.2

−0.500000

+

0.866025i

1.00000i

−0.500000

−

0.866025i

0.500000

+

0.866025i

−0.866025

−

0.500000i

0

1.00000

−1.00000

−1.00000

1039.1

−0.500000

−

0.866025i

−

1.00000i

−0.500000

+

0.866025i

0.500000

−

0.866025i

−0.866025

+

0.500000i

0

1.00000

−1.00000

−1.00000

1039.2

−0.500000

−

0.866025i

1.00000i

−0.500000

+

0.866025i

0.500000

−

0.866025i

0.866025

−

0.500000i

0

1.00000

−1.00000

−1.00000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
260.g	odd	2	1	CM by \(\Q(\sqrt{-65}) \)
4.b	odd	2	1	inner
9.c	even	3	1	inner
36.f	odd	6	1	inner
65.d	even	2	1	inner
585.be	even	6	1	inner
2340.db	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2340.1.db.e	✓	4
4.b	odd	2	1	inner	2340.1.db.e	✓	4
5.b	even	2	1		2340.1.db.f	yes	4
9.c	even	3	1	inner	2340.1.db.e	✓	4
13.b	even	2	1		2340.1.db.f	yes	4
20.d	odd	2	1		2340.1.db.f	yes	4
36.f	odd	6	1	inner	2340.1.db.e	✓	4
45.j	even	6	1		2340.1.db.f	yes	4
52.b	odd	2	1		2340.1.db.f	yes	4
65.d	even	2	1	inner	2340.1.db.e	✓	4
117.t	even	6	1		2340.1.db.f	yes	4
180.p	odd	6	1		2340.1.db.f	yes	4
260.g	odd	2	1	CM	2340.1.db.e	✓	4
468.bg	odd	6	1		2340.1.db.f	yes	4
585.be	even	6	1	inner	2340.1.db.e	✓	4
2340.db	odd	6	1	inner	2340.1.db.e	✓	4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
2340.1.db.e	✓	4	1.a	even	1	1	trivial
2340.1.db.e	✓	4	4.b	odd	2	1	inner
2340.1.db.e	✓	4	9.c	even	3	1	inner
2340.1.db.e	✓	4	36.f	odd	6	1	inner
2340.1.db.e	✓	4	65.d	even	2	1	inner
2340.1.db.e	✓	4	260.g	odd	2	1	CM
2340.1.db.e	✓	4	585.be	even	6	1	inner
2340.1.db.e	✓	4	2340.db	odd	6	1	inner
2340.1.db.f	yes	4	5.b	even	2	1
2340.1.db.f	yes	4	13.b	even	2	1
2340.1.db.f	yes	4	20.d	odd	2	1
2340.1.db.f	yes	4	45.j	even	6	1
2340.1.db.f	yes	4	52.b	odd	2	1
2340.1.db.f	yes	4	117.t	even	6	1
2340.1.db.f	yes	4	180.p	odd	6	1
2340.1.db.f	yes	4	468.bg	odd	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(2340, [\chi])\):

\( T_{11}^{4} + 3T_{11}^{2} + 9 \)

\( T_{23} \)

\( T_{37} + 1 \)

Hecke characteristic polynomials

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