LMFDB - Newform orbit 2380.1.bg.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2380,1,Mod(489,2380)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2380.489"); S:= CuspForms(chi, 1); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2380, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 2, 2, 1])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]

Level:	$ N $	$=$	$ 2380 = 2^{2} \cdot 5 \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2380.bg (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$1.18777473007$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
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:	$\Z[a_1, \ldots, a_{17}]$
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}^{11} + \zeta_{24}^{7}) q^{3} + \zeta_{24}^{9} q^{5} - \zeta_{24}^{3} q^{7} + ( - \zeta_{24}^{10} + \cdots - \zeta_{24}^{2}) q^{9} + (\zeta_{24}^{4} + \zeta_{24}^{2}) q^{11} + ( - \zeta_{24}^{11} + \zeta_{24}) q^{13} + \cdots + ( - \zeta_{24}^{10} - \zeta_{24}^{8} + \cdots + 1) q^{99} +O(q^{100}) $ q + (z^11 + z^7) * q^3 + z^9 * q^5 - z^3 * q^7 + (-z^10 - z^6 - z^2) * q^9 + (z^4 + z^2) * q^11 + (-z^11 + z) * q^13 + (-z^8 - z^4) * q^15 + z * q^17 + (-z^10 + z^2) * q^21 - z^6 * q^25 + (z^5 + z) * q^27 + (z^4 - z^2) * q^29 + (z^11 + z^9 - z^3 - z) * q^33 + q^35 + (z^10 + z^8 + z^6 - 1) * q^39 + (-z^11 + z^7 + z^3) * q^45 + (-z^7 + z^5) * q^47 + z^6 * q^49 + (z^8 - 1) * q^51 + (z^11 - z) * q^55 + (z^9 + z^5 - z) * q^63 + (z^10 + z^8) * q^65 + (z^6 - 1) * q^71 - 2z^9 q^73 + (z^5 + z) * q^75 + (-z^7 - z^5) * q^77 + (z^10 - z^8) * q^79 - q^81 + (z^9 + z^3) * q^83 + z^10 * q^85 + (z^11 - z^9 - z^3 + z) * q^87 + (-z^4 - z^2) * q^91 + (z^11 + z^7) * q^97 + (-z^10 - z^8 - z^6 - z^4 + z^2 + 1) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{11} + 4 q^{29} + 8 q^{35} - 12 q^{39} - 12 q^{51} - 4 q^{65} - 8 q^{71} + 4 q^{79} - 8 q^{81} - 4 q^{91} + 8 q^{99}+O(q^{100}) $ 8 * q + 4 * q^11 + 4 * q^29 + 8 * q^35 - 12 * q^39 - 12 * q^51 - 4 * q^65 - 8 * q^71 + 4 * q^79 - 8 * q^81 - 4 * q^91 + 8 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2380\mathbb{Z}\right)^\times$.

$n$	$477$	$1191$	$1261$	$1361$
$\chi(n)$	$-1$	$1$	$-\zeta_{24}^{6}$	$-1$

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} $

489.1

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
−0.965926	−	0.258819i

−1.22474

−

1.22474i

−0.707107

−

0.707107i

−0.707107

0.707107i

2.00000i

489.2

−1.22474

−

1.22474i

0.707107

0.707107i

0.707107

−

0.707107i

2.00000i

489.3

1.22474

1.22474i

−0.707107

−

0.707107i

−0.707107

0.707107i

2.00000i

489.4

1.22474

1.22474i

0.707107

0.707107i

0.707107

−

0.707107i

2.00000i

769.1

−1.22474

1.22474i

−0.707107

0.707107i

−0.707107

−

0.707107i

−

2.00000i

769.2

−1.22474

1.22474i

0.707107

−

0.707107i

0.707107

0.707107i

−

2.00000i

769.3

1.22474

−

1.22474i

−0.707107

0.707107i

−0.707107

−

0.707107i

−

2.00000i

769.4

1.22474

−

1.22474i

0.707107

−

0.707107i

0.707107

0.707107i

−

2.00000i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
35.c	odd	2	1	CM by $\Q(\sqrt{-35}) $
5.b	even	2	1	inner
7.b	odd	2	1	inner
17.c	even	4	1	inner
85.j	even	4	1	inner
119.f	odd	4	1	inner
595.u	odd	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2380.1.bg.a	✓	8
5.b	even	2	1	inner	2380.1.bg.a	✓	8
7.b	odd	2	1	inner	2380.1.bg.a	✓	8
17.c	even	4	1	inner	2380.1.bg.a	✓	8
35.c	odd	2	1	CM	2380.1.bg.a	✓	8
85.j	even	4	1	inner	2380.1.bg.a	✓	8
119.f	odd	4	1	inner	2380.1.bg.a	✓	8
595.u	odd	4	1	inner	2380.1.bg.a	✓	8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
2380.1.bg.a	✓	8	1.a	even	1	1	trivial
2380.1.bg.a	✓	8	5.b	even	2	1	inner
2380.1.bg.a	✓	8	7.b	odd	2	1	inner
2380.1.bg.a	✓	8	17.c	even	4	1	inner
2380.1.bg.a	✓	8	35.c	odd	2	1	CM
2380.1.bg.a	✓	8	85.j	even	4	1	inner
2380.1.bg.a	✓	8	119.f	odd	4	1	inner
2380.1.bg.a	✓	8	595.u	odd	4	1	inner

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(2380, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{8} $ T^8
$3$	$ (T^{4} + 9)^{2} $ (T^4 + 9)^2
$5$	$ (T^{4} + 1)^{2} $ (T^4 + 1)^2
$7$	$ (T^{4} + 1)^{2} $ (T^4 + 1)^2
$11$	$ (T^{4} - 2 T^{3} + 2 T^{2} + \cdots + 1)^{2} $ (T^4 - 2T^3 + 2T^2 + 2*T + 1)^2
$13$	$ (T^{4} - 4 T^{2} + 1)^{2} $ (T^4 - 4*T^2 + 1)^2
$17$	$ T^{8} - T^{4} + 1 $ T^8 - T^4 + 1
$19$	$ T^{8} $ T^8
$23$	$ T^{8} $ T^8
$29$	$ (T^{4} - 2 T^{3} + 2 T^{2} + \cdots + 1)^{2} $ (T^4 - 2T^3 + 2T^2 + 2*T + 1)^2
$31$	$ T^{8} $ T^8
$37$	$ T^{8} $ T^8
$41$	$ T^{8} $ T^8
$43$	$ T^{8} $ T^8
$47$	$ (T^{4} - 4 T^{2} + 1)^{2} $ (T^4 - 4*T^2 + 1)^2
$53$	$ T^{8} $ T^8
$59$	$ T^{8} $ T^8
$61$	$ T^{8} $ T^8
$67$	$ T^{8} $ T^8
$71$	$ (T^{2} + 2 T + 2)^{4} $ (T^2 + 2*T + 2)^4
$73$	$ (T^{4} + 16)^{2} $ (T^4 + 16)^2
$79$	$ (T^{4} - 2 T^{3} + 2 T^{2} + \cdots + 1)^{2} $ (T^4 - 2T^3 + 2T^2 + 2*T + 1)^2
$83$	$ (T^{2} + 2)^{4} $ (T^2 + 2)^4
$89$	$ T^{8} $ T^8
$97$	$ (T^{4} + 9)^{2} $ (T^4 + 9)^2
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Level:	\( N \)	\(=\)	\( 2380 = 2^{2} \cdot 5 \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2380.bg (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\( q + (\zeta_{24}^{11} + \zeta_{24}^{7}) q^{3} + \zeta_{24}^{9} q^{5} - \zeta_{24}^{3} q^{7} + ( - \zeta_{24}^{10} + \cdots - \zeta_{24}^{2}) q^{9} + (\zeta_{24}^{4} + \zeta_{24}^{2}) q^{11} + ( - \zeta_{24}^{11} + \zeta_{24}) q^{13} + \cdots + ( - \zeta_{24}^{10} - \zeta_{24}^{8} + \cdots + 1) q^{99} +O(q^{100}) \) q + (z^11 + z^7) * q^3 + z^9 * q^5 - z^3 * q^7 + (-z^10 - z^6 - z^2) * q^9 + (z^4 + z^2) * q^11 + (-z^11 + z) * q^13 + (-z^8 - z^4) * q^15 + z * q^17 + (-z^10 + z^2) * q^21 - z^6 * q^25 + (z^5 + z) * q^27 + (z^4 - z^2) * q^29 + (z^11 + z^9 - z^3 - z) * q^33 + q^35 + (z^10 + z^8 + z^6 - 1) * q^39 + (-z^11 + z^7 + z^3) * q^45 + (-z^7 + z^5) * q^47 + z^6 * q^49 + (z^8 - 1) * q^51 + (z^11 - z) * q^55 + (z^9 + z^5 - z) * q^63 + (z^10 + z^8) * q^65 + (z^6 - 1) * q^71 - 2z^9 q^73 + (z^5 + z) * q^75 + (-z^7 - z^5) * q^77 + (z^10 - z^8) * q^79 - q^81 + (z^9 + z^3) * q^83 + z^10 * q^85 + (z^11 - z^9 - z^3 + z) * q^87 + (-z^4 - z^2) * q^91 + (z^11 + z^7) * q^97 + (-z^10 - z^8 - z^6 - z^4 + z^2 + 1) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{11} + 4 q^{29} + 8 q^{35} - 12 q^{39} - 12 q^{51} - 4 q^{65} - 8 q^{71} + 4 q^{79} - 8 q^{81} - 4 q^{91} + 8 q^{99}+O(q^{100}) \) 8 * q + 4 * q^11 + 4 * q^29 + 8 * q^35 - 12 * q^39 - 12 * q^51 - 4 * q^65 - 8 * q^71 + 4 * q^79 - 8 * q^81 - 4 * q^91 + 8 * q^99

Introduction

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Varieties

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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	2380.1.bg.a

Level	$2380$
Weight	$1$
Character orbit	2380.bg
Analytic conductor	$1.188$
Analytic rank	$0$
Dimension	$8$
Projective image	$D_{12}$
CM discriminant	-35
Inner twists	$8$

Self dual:	no
Analytic conductor:	\(1.18777473007\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(4\) over \(\Q(i)\)
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:	\(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{12}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{12} - \cdots)\)

\(n\)	\(477\)	\(1191\)	\(1261\)	\(1361\)
\(\chi(n)\)	\(-1\)	\(1\)	\(-\zeta_{24}^{6}\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 489.4
Significant digits:
Format:

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