LMFDB - Newform orbit 2420.1.j.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2420,1,Mod(1453,2420)] 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("2420.1453"); S:= CuspForms(chi, 1); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2420, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 3, 0])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]

Level:	$ N $	$=$	$ 2420 = 2^{2} \cdot 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2420.j (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:	$1.20773733057$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(\zeta_{12})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{12}$
Projective field:	Galois closure of 12.2.80525500000000.1

$q$-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_{12}^{2} - \zeta_{12}) q^{3} + \zeta_{12}^{5} q^{5} + (\zeta_{12}^{4} + \cdots + \zeta_{12}^{2}) q^{9} + (\zeta_{12} + 1) q^{15} + (\zeta_{12}^{5} - \zeta_{12}^{4}) q^{23} - \zeta_{12}^{4} q^{25} + \cdots + ( - \zeta_{12}^{5} - \zeta_{12}^{4}) q^{97} +O(q^{100}) $ q + (-z^2 - z) * q^3 + z^5 * q^5 + (z^4 + z^3 + z^2) * q^9 + (z + 1) * q^15 + (z^5 - z^4) * q^23 - z^4 * q^25 + (-z^5 - z^4 - z^3 + 1) * q^27 + (-z^5 + z) * q^31 + (z^2 - z) * q^37 + (-z^3 - z^2 - z) * q^45 + (-z^3 + 1) * q^47 + z^3 * q^49 + (-z^3 - 1) * q^53 + (-z^4 - z^2) * q^59 + (-z^2 + z) * q^67 + (z^5 + z) * q^69 - q^71 + (z^5 - 1) * q^75 + (z^5 + z^4 - z^2 - z - 1) * q^81 - z^3 * q^89 + (-z^3 - z^2 - z - 1) * q^93 + (-z^5 - z^4) * q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{3} + 4 q^{15} + 2 q^{23} + 2 q^{25} + 6 q^{27} + 2 q^{37} - 2 q^{45} + 4 q^{47} - 4 q^{53} - 2 q^{67} - 4 q^{71} - 4 q^{75} - 8 q^{81} - 6 q^{93} + 2 q^{97}+O(q^{100}) $ 4 * q - 2 * q^3 + 4 * q^15 + 2 * q^23 + 2 * q^25 + 6 * q^27 + 2 * q^37 - 2 * q^45 + 4 * q^47 - 4 * q^53 - 2 * q^67 - 4 * q^71 - 4 * q^75 - 8 * q^81 - 6 * q^93 + 2 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2420\mathbb{Z}\right)^\times$.

$n$	$1211$	$1937$	$2301$
$\chi(n)$	$1$	$-\zeta_{12}^{3}$	$1$

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} $

1453.1

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

−1.36603

−

1.36603i

−0.866025

0.500000i

2.73205i

1453.2

0.366025

0.366025i

0.866025

0.500000i

−

0.732051i

1937.1

−1.36603

1.36603i

−0.866025

−

0.500000i

−

2.73205i

1937.2

0.366025

−

0.366025i

0.866025

−

0.500000i

0.732051i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
11.b	odd	2	1	CM by $\Q(\sqrt{-11}) $
5.c	odd	4	1	inner
55.e	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2420.1.j.a	✓	4
5.c	odd	4	1	inner	2420.1.j.a	✓	4
11.b	odd	2	1	CM	2420.1.j.a	✓	4
11.c	even	5	4		2420.1.y.a		16
11.d	odd	10	4		2420.1.y.a		16
55.e	even	4	1	inner	2420.1.j.a	✓	4
55.k	odd	20	4		2420.1.y.a		16
55.l	even	20	4		2420.1.y.a		16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
2420.1.j.a	✓	4	1.a	even	1	1	trivial
2420.1.j.a	✓	4	5.c	odd	4	1	inner
2420.1.j.a	✓	4	11.b	odd	2	1	CM
2420.1.j.a	✓	4	55.e	even	4	1	inner
2420.1.y.a		16	11.c	even	5	4
2420.1.y.a		16	11.d	odd	10	4
2420.1.y.a		16	55.k	odd	20	4
2420.1.y.a		16	55.l	even	20	4

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(2420, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{4} $ T^4
$3$	$ T^{4} + 2 T^{3} + \cdots + 1 $ T^4 + 2T^3 + 2T^2 - 2*T + 1
$5$	$ T^{4} - T^{2} + 1 $ T^4 - T^2 + 1
$7$	$ T^{4} $ T^4
$11$	$ T^{4} $ T^4
$13$	$ T^{4} $ T^4
$17$	$ T^{4} $ T^4
$19$	$ T^{4} $ T^4
$23$	$ T^{4} - 2 T^{3} + \cdots + 1 $ T^4 - 2T^3 + 2T^2 + 2*T + 1
$29$	$ T^{4} $ T^4
$31$	$ (T^{2} - 3)^{2} $ (T^2 - 3)^2
$37$	$ T^{4} - 2 T^{3} + \cdots + 1 $ T^4 - 2T^3 + 2T^2 + 2*T + 1
$41$	$ T^{4} $ T^4
$43$	$ T^{4} $ T^4
$47$	$ (T^{2} - 2 T + 2)^{2} $ (T^2 - 2*T + 2)^2
$53$	$ (T^{2} + 2 T + 2)^{2} $ (T^2 + 2*T + 2)^2
$59$	$ (T^{2} + 3)^{2} $ (T^2 + 3)^2
$61$	$ T^{4} $ T^4
$67$	$ T^{4} + 2 T^{3} + \cdots + 1 $ T^4 + 2T^3 + 2T^2 - 2*T + 1
$71$	$ (T + 1)^{4} $ (T + 1)^4
$73$	$ T^{4} $ T^4
$79$	$ T^{4} $ T^4
$83$	$ T^{4} $ T^4
$89$	$ (T^{2} + 1)^{2} $ (T^2 + 1)^2
$97$	$ T^{4} - 2 T^{3} + \cdots + 1 $ T^4 - 2T^3 + 2T^2 + 2*T + 1
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Level:	\( N \)	\(=\)	\( 2420 = 2^{2} \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2420.j (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\( q + ( - \zeta_{12}^{2} - \zeta_{12}) q^{3} + \zeta_{12}^{5} q^{5} + (\zeta_{12}^{4} + \cdots + \zeta_{12}^{2}) q^{9} + (\zeta_{12} + 1) q^{15} + (\zeta_{12}^{5} - \zeta_{12}^{4}) q^{23} - \zeta_{12}^{4} q^{25} + \cdots + ( - \zeta_{12}^{5} - \zeta_{12}^{4}) q^{97} +O(q^{100}) \) q + (-z^2 - z) * q^3 + z^5 * q^5 + (z^4 + z^3 + z^2) * q^9 + (z + 1) * q^15 + (z^5 - z^4) * q^23 - z^4 * q^25 + (-z^5 - z^4 - z^3 + 1) * q^27 + (-z^5 + z) * q^31 + (z^2 - z) * q^37 + (-z^3 - z^2 - z) * q^45 + (-z^3 + 1) * q^47 + z^3 * q^49 + (-z^3 - 1) * q^53 + (-z^4 - z^2) * q^59 + (-z^2 + z) * q^67 + (z^5 + z) * q^69 - q^71 + (z^5 - 1) * q^75 + (z^5 + z^4 - z^2 - z - 1) * q^81 - z^3 * q^89 + (-z^3 - z^2 - z - 1) * q^93 + (-z^5 - z^4) * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{3} + 4 q^{15} + 2 q^{23} + 2 q^{25} + 6 q^{27} + 2 q^{37} - 2 q^{45} + 4 q^{47} - 4 q^{53} - 2 q^{67} - 4 q^{71} - 4 q^{75} - 8 q^{81} - 6 q^{93} + 2 q^{97}+O(q^{100}) \) 4 * q - 2 * q^3 + 4 * q^15 + 2 * q^23 + 2 * q^25 + 6 * q^27 + 2 * q^37 - 2 * q^45 + 4 * q^47 - 4 * q^53 - 2 * q^67 - 4 * q^71 - 4 * q^75 - 8 * q^81 - 6 * q^93 + 2 * q^97

Introduction

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Varieties

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Representations

Groups

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	2420.1.j.a

Level	$2420$
Weight	$1$
Character orbit	2420.j
Analytic conductor	$1.208$
Analytic rank	$0$
Dimension	$4$
Projective image	$D_{12}$
CM discriminant	-11
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(1.20773733057\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Relative dimension:	\(2\) over \(\Q(i)\)
Coefficient field:	\(\Q(\zeta_{12})\)
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{4} - x^{2} + 1 \) x^4 - x^2 + 1
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{12}\)
Projective field:	Galois closure of 12.2.80525500000000.1

\(n\)	\(1211\)	\(1937\)	\(2301\)
\(\chi(n)\)	\(1\)	\(-\zeta_{12}^{3}\)	\(1\)

$p$	$F_p(T)$
$2$	\( T^{4} \) T^4
$3$	\( T^{4} + 2 T^{3} + \cdots + 1 \) T^4 + 2T^3 + 2T^2 - 2*T + 1
$5$	\( T^{4} - T^{2} + 1 \) T^4 - T^2 + 1
$7$	\( T^{4} \) T^4
$11$	\( T^{4} \) T^4
$13$	\( T^{4} \) T^4
$17$	\( T^{4} \) T^4
$19$	\( T^{4} \) T^4
$23$	\( T^{4} - 2 T^{3} + \cdots + 1 \) T^4 - 2T^3 + 2T^2 + 2*T + 1
$29$	\( T^{4} \) T^4
$31$	\( (T^{2} - 3)^{2} \) (T^2 - 3)^2
$37$	\( T^{4} - 2 T^{3} + \cdots + 1 \) T^4 - 2T^3 + 2T^2 + 2*T + 1
$41$	\( T^{4} \) T^4
$43$	\( T^{4} \) T^4
$47$	\( (T^{2} - 2 T + 2)^{2} \) (T^2 - 2*T + 2)^2
$53$	\( (T^{2} + 2 T + 2)^{2} \) (T^2 + 2*T + 2)^2
$59$	\( (T^{2} + 3)^{2} \) (T^2 + 3)^2
$61$	\( T^{4} \) T^4
$67$	\( T^{4} + 2 T^{3} + \cdots + 1 \) T^4 + 2T^3 + 2T^2 - 2*T + 1
$71$	\( (T + 1)^{4} \) (T + 1)^4
$73$	\( T^{4} \) T^4
$79$	\( T^{4} \) T^4
$83$	\( T^{4} \) T^4
$89$	\( (T^{2} + 1)^{2} \) (T^2 + 1)^2
$97$	\( T^{4} - 2 T^{3} + \cdots + 1 \) T^4 - 2T^3 + 2T^2 + 2*T + 1
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\(n\):		e.g. 2-40 or 990-1000
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