LMFDB - Newform orbit 2340.1.ga.b

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2340,1,Mod(539,2340)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2340, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([6, 6, 6, 5])) N = Newforms(chi, 1, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2340.539"); S:= CuspForms(chi, 1); N := Newforms(S);

Level:	$ N $	$=$	$ 2340 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2340.ga (of order $12$, degree $4$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,0,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(10)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$1.16781212956$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$2$ over $\Q(\zeta_{12})$
Coefficient field:	$\Q(\zeta_{24})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{12}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{12} - \cdots)$

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

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

$f(q)$	$=$	$ q + \zeta_{24}^{5} q^{2} + \zeta_{24}^{10} q^{4} - \zeta_{24}^{7} q^{5} - \zeta_{24}^{3} q^{8} + q^{10} + \zeta_{24}^{8} q^{13} - \zeta_{24}^{8} q^{16} + ( - \zeta_{24}^{11} - \zeta_{24}^{9}) q^{17} + \zeta_{24}^{5} q^{20} + \cdots + \zeta_{24}^{7} q^{98} +O(q^{100}) $ q + z^5 * q^2 + z^10 * q^4 - z^7 * q^5 - z^3 * q^8 + q^10 + z^8 * q^13 - z^8 * q^16 + (-z^11 - z^9) * q^17 + z^5 * q^20 - z^2 * q^25 - z * q^26 + (z^3 + z) * q^29 + z * q^32 + (z^4 + z^2) * q^34 + (-z^6 - z^4) * q^37 + z^10 * q^40 + (-z^9 + z) * q^41 + z^2 * q^49 - z^7 * q^50 - z^6 * q^52 + (z^7 - z^5) * q^53 + (z^8 + z^6) * q^58 + z^4 * q^61 + z^6 * q^64 + z^3 * q^65 + (z^9 + z^7) * q^68 + (z^4 - z^2) * q^73 + (-z^11 - z^9) * q^74 - z^3 * q^80 + (z^6 + z^2) * q^82 + (-z^6 - z^4) * q^85 - 2z^5 q^89 + (z^10 + z^4) * q^97 + z^7 * q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{10} - 4 q^{13} + 4 q^{16} + 4 q^{34} - 4 q^{37} - 4 q^{58} + 4 q^{61} + 4 q^{73} - 4 q^{85} + 4 q^{97}+O(q^{100}) $ 8 * q + 8 * q^10 - 4 * q^13 + 4 * q^16 + 4 * q^34 - 4 * q^37 - 4 * q^58 + 4 * q^61 + 4 * q^73 - 4 * q^85 + 4 * q^97

Character values

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

$n$	$937$	$1081$	$1171$	$2081$
$\chi(n)$	$-1$	$\zeta_{24}^{10}$	$-1$	$-1$

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.

comment:embeddings in the coefficient field

gp:mfembed(f)

Label

$\iota_m(\nu)$

$ a_{2} $

$ a_{3} $

$ a_{4} $

$ a_{5} $

$ a_{6} $

$ a_{7} $

$ a_{8} $

$ a_{9} $

$ a_{10} $

539.1

−0.965926	−	0.258819i
0.965926	+	0.258819i
−0.258819	−	0.965926i
0.258819	+	0.965926i
−0.965926	+	0.258819i
0.965926	−	0.258819i
−0.258819	+	0.965926i
0.258819	−	0.965926i

−0.258819

−

0.965926i

−0.866025

0.500000i

−0.258819

0.965926i

0.707107

0.707107i

1.00000

539.2

0.258819

0.965926i

−0.866025

0.500000i

0.258819

−

0.965926i

−0.707107

−

0.707107i

1.00000

899.1

−0.965926

−

0.258819i

0.866025

0.500000i

−0.965926

0.258819i

−0.707107

−

0.707107i

1.00000

899.2

0.965926

0.258819i

0.866025

0.500000i

0.965926

−

0.258819i

0.707107

0.707107i

1.00000

1259.1

−0.258819

0.965926i

−0.866025

−

0.500000i

−0.258819

−

0.965926i

0.707107

−

0.707107i

1.00000

1259.2

0.258819

−

0.965926i

−0.866025

−

0.500000i

0.258819

0.965926i

−0.707107

0.707107i

1.00000

1619.1

−0.965926

0.258819i

0.866025

−

0.500000i

−0.965926

−

0.258819i

−0.707107

0.707107i

1.00000

1619.2

0.965926

−

0.258819i

0.866025

−

0.500000i

0.965926

0.258819i

0.707107

−

0.707107i

1.00000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	CM by $\Q(\sqrt{-1}) $
3.b	odd	2	1	inner
12.b	even	2	1	inner
65.s	odd	12	1	inner
195.bh	even	12	1	inner
260.bc	even	12	1	inner
780.cr	odd	12	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2340.1.ga.b	yes	8
3.b	odd	2	1	inner	2340.1.ga.b	yes	8
4.b	odd	2	1	CM	2340.1.ga.b	yes	8
5.b	even	2	1		2340.1.ga.a	✓	8
12.b	even	2	1	inner	2340.1.ga.b	yes	8
13.f	odd	12	1		2340.1.ga.a	✓	8
15.d	odd	2	1		2340.1.ga.a	✓	8
20.d	odd	2	1		2340.1.ga.a	✓	8
39.k	even	12	1		2340.1.ga.a	✓	8
52.l	even	12	1		2340.1.ga.a	✓	8
60.h	even	2	1		2340.1.ga.a	✓	8
65.s	odd	12	1	inner	2340.1.ga.b	yes	8
156.v	odd	12	1		2340.1.ga.a	✓	8
195.bh	even	12	1	inner	2340.1.ga.b	yes	8
260.bc	even	12	1	inner	2340.1.ga.b	yes	8
780.cr	odd	12	1	inner	2340.1.ga.b	yes	8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
2340.1.ga.a	✓	8	5.b	even	2	1
2340.1.ga.a	✓	8	13.f	odd	12	1
2340.1.ga.a	✓	8	15.d	odd	2	1
2340.1.ga.a	✓	8	20.d	odd	2	1
2340.1.ga.a	✓	8	39.k	even	12	1
2340.1.ga.a	✓	8	52.l	even	12	1
2340.1.ga.a	✓	8	60.h	even	2	1
2340.1.ga.a	✓	8	156.v	odd	12	1
2340.1.ga.b	yes	8	1.a	even	1	1	trivial
2340.1.ga.b	yes	8	3.b	odd	2	1	inner
2340.1.ga.b	yes	8	4.b	odd	2	1	CM
2340.1.ga.b	yes	8	12.b	even	2	1	inner
2340.1.ga.b	yes	8	65.s	odd	12	1	inner
2340.1.ga.b	yes	8	195.bh	even	12	1	inner
2340.1.ga.b	yes	8	260.bc	even	12	1	inner
2340.1.ga.b	yes	8	780.cr	odd	12	1	inner

This newform subspace can be constructed as the kernel of the linear operator $ T_{37}^{4} + 2T_{37}^{3} + 5T_{37}^{2} + 4T_{37} + 1 $ T37^4 + 2*T37^3 + 5*T37^2 + 4*T37 + 1 acting on $S_{1}^{\mathrm{new}}(2340, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{8} - T^{4} + 1 $ T^8 - T^4 + 1
$3$	$ T^{8} $ T^8
$5$	$ T^{8} - T^{4} + 1 $ T^8 - T^4 + 1
$7$	$ T^{8} $ T^8
$11$	$ T^{8} $ T^8
$13$	$ (T^{2} + T + 1)^{4} $ (T^2 + T + 1)^4
$17$	$ T^{8} - 4 T^{6} + \cdots + 1 $ T^8 - 4T^6 + 15T^4 - 4*T^2 + 1
$19$	$ T^{8} $ T^8
$23$	$ T^{8} $ T^8
$29$	$ T^{8} - 4 T^{6} + \cdots + 1 $ T^8 - 4T^6 + 15T^4 - 4*T^2 + 1
$31$	$ T^{8} $ T^8
$37$	$ (T^{4} + 2 T^{3} + 5 T^{2} + \cdots + 1)^{2} $ (T^4 + 2T^3 + 5T^2 + 4*T + 1)^2
$41$	$ T^{8} - 9T^{4} + 81 $ T^8 - 9*T^4 + 81
$43$	$ T^{8} $ T^8
$47$	$ T^{8} $ T^8
$53$	$ (T^{4} - 4 T^{2} + 1)^{2} $ (T^4 - 4*T^2 + 1)^2
$59$	$ T^{8} $ T^8
$61$	$ (T^{2} - T + 1)^{4} $ (T^2 - T + 1)^4
$67$	$ T^{8} $ T^8
$71$	$ T^{8} $ T^8
$73$	$ (T^{4} - 2 T^{3} + 2 T^{2} + \cdots + 1)^{2} $ (T^4 - 2T^3 + 2T^2 + 2*T + 1)^2
$79$	$ T^{8} $ T^8
$83$	$ T^{8} $ T^8
$89$	$ T^{8} - 16T^{4} + 256 $ T^8 - 16*T^4 + 256
$97$	$ (T^{4} - 2 T^{3} + 2 T^{2} + \cdots + 4)^{2} $ (T^4 - 2T^3 + 2T^2 - 4*T + 4)^2
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Level:	\( N \)	\(=\)	\( 2340 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2340.ga (of order \(12\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\( q + \zeta_{24}^{5} q^{2} + \zeta_{24}^{10} q^{4} - \zeta_{24}^{7} q^{5} - \zeta_{24}^{3} q^{8} + q^{10} + \zeta_{24}^{8} q^{13} - \zeta_{24}^{8} q^{16} + ( - \zeta_{24}^{11} - \zeta_{24}^{9}) q^{17} + \zeta_{24}^{5} q^{20} + \cdots + \zeta_{24}^{7} q^{98} +O(q^{100}) \) q + z^5 * q^2 + z^10 * q^4 - z^7 * q^5 - z^3 * q^8 + q^10 + z^8 * q^13 - z^8 * q^16 + (-z^11 - z^9) * q^17 + z^5 * q^20 - z^2 * q^25 - z * q^26 + (z^3 + z) * q^29 + z * q^32 + (z^4 + z^2) * q^34 + (-z^6 - z^4) * q^37 + z^10 * q^40 + (-z^9 + z) * q^41 + z^2 * q^49 - z^7 * q^50 - z^6 * q^52 + (z^7 - z^5) * q^53 + (z^8 + z^6) * q^58 + z^4 * q^61 + z^6 * q^64 + z^3 * q^65 + (z^9 + z^7) * q^68 + (z^4 - z^2) * q^73 + (-z^11 - z^9) * q^74 - z^3 * q^80 + (z^6 + z^2) * q^82 + (-z^6 - z^4) * q^85 - 2z^5 q^89 + (z^10 + z^4) * q^97 + z^7 * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{10} - 4 q^{13} + 4 q^{16} + 4 q^{34} - 4 q^{37} - 4 q^{58} + 4 q^{61} + 4 q^{73} - 4 q^{85} + 4 q^{97}+O(q^{100}) \) 8 * q + 8 * q^10 - 4 * q^13 + 4 * q^16 + 4 * q^34 - 4 * q^37 - 4 * q^58 + 4 * q^61 + 4 * q^73 - 4 * q^85 + 4 * q^97

Introduction

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Varieties

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Database

Properties

Related objects

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Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	2340.1.ga.b

Level	$2340$
Weight	$1$
Character orbit	2340.ga
Analytic conductor	$1.168$
Analytic rank	$0$
Dimension	$8$
Projective image	$D_{12}$
CM discriminant	-4
Inner twists	$8$

Self dual:	no
Analytic conductor:	\(1.16781212956\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(2\) over \(\Q(\zeta_{12})\)
Coefficient field:	\(\Q(\zeta_{24})\)
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{8} - x^{4} + 1 \) x^8 - x^4 + 1
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{12}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{12} - \cdots)\)

\(n\)	\(937\)	\(1081\)	\(1171\)	\(2081\)
\(\chi(n)\)	\(-1\)	\(\zeta_{24}^{10}\)	\(-1\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 539.2
Significant digits:
Format:

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	CM by \(\Q(\sqrt{-1}) \)
3.b	odd	2	1	inner
12.b	even	2	1	inner
65.s	odd	12	1	inner
195.bh	even	12	1	inner
260.bc	even	12	1	inner
780.cr	odd	12	1	inner

$p$	$F_p(T)$
$2$	\( T^{8} - T^{4} + 1 \) T^8 - T^4 + 1
$3$	\( T^{8} \) T^8
$5$	\( T^{8} - T^{4} + 1 \) T^8 - T^4 + 1
$7$	\( T^{8} \) T^8
$11$	\( T^{8} \) T^8
$13$	\( (T^{2} + T + 1)^{4} \) (T^2 + T + 1)^4
$17$	\( T^{8} - 4 T^{6} + \cdots + 1 \) T^8 - 4T^6 + 15T^4 - 4*T^2 + 1
$19$	\( T^{8} \) T^8
$23$	\( T^{8} \) T^8
$29$	\( T^{8} - 4 T^{6} + \cdots + 1 \) T^8 - 4T^6 + 15T^4 - 4*T^2 + 1
$31$	\( T^{8} \) T^8
$37$	\( (T^{4} + 2 T^{3} + 5 T^{2} + \cdots + 1)^{2} \) (T^4 + 2T^3 + 5T^2 + 4*T + 1)^2
$41$	\( T^{8} - 9T^{4} + 81 \) T^8 - 9*T^4 + 81
$43$	\( T^{8} \) T^8
$47$	\( T^{8} \) T^8
$53$	\( (T^{4} - 4 T^{2} + 1)^{2} \) (T^4 - 4*T^2 + 1)^2
$59$	\( T^{8} \) T^8
$61$	\( (T^{2} - T + 1)^{4} \) (T^2 - T + 1)^4
$67$	\( T^{8} \) T^8
$71$	\( T^{8} \) T^8
$73$	\( (T^{4} - 2 T^{3} + 2 T^{2} + \cdots + 1)^{2} \) (T^4 - 2T^3 + 2T^2 + 2*T + 1)^2
$79$	\( T^{8} \) T^8
$83$	\( T^{8} \) T^8
$89$	\( T^{8} - 16T^{4} + 256 \) T^8 - 16*T^4 + 256
$97$	\( (T^{4} - 2 T^{3} + 2 T^{2} + \cdots + 4)^{2} \) (T^4 - 2T^3 + 2T^2 - 4*T + 4)^2
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