LMFDB - Newform orbit 3900.1.dp.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3900,1,Mod(293,3900)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3900, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([0, 6, 9, 11]))

N = Newforms(chi, 1, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3900.293");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3900.dp (of order $12$, degree $4$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$1.94635354927$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{12}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{12} - \cdots)$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3900\mathbb{Z}\right)^\times$.

$n$	$301$	$1301$	$1951$	$3277$
$\chi(n)$	$\zeta_{24}^{10}$	$-1$	$1$	$\zeta_{24}^{6}$

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} $

293.1

−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
0.965926	+	0.258819i

−0.965926

0.258819i

1.67303

−

0.965926i

0.866025

−

0.500000i

293.2

0.965926

−

0.258819i

−1.67303

0.965926i

0.866025

−

0.500000i

1493.1

−0.258819

0.965926i

−0.448288

−

0.258819i

−0.866025

−

0.500000i

1493.2

0.258819

−

0.965926i

0.448288

0.258819i

−0.866025

−

0.500000i

1757.1

−0.965926

−

0.258819i

1.67303

0.965926i

0.866025

0.500000i

1757.2

0.965926

0.258819i

−1.67303

−

0.965926i

0.866025

0.500000i

2957.1

−0.258819

−

0.965926i

−0.448288

0.258819i

−0.866025

0.500000i

2957.2

0.258819

0.965926i

0.448288

−

0.258819i

−0.866025

0.500000i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by $\Q(\sqrt{-3}) $
5.b	even	2	1	inner
15.d	odd	2	1	inner
65.o	even	12	1	inner
65.t	even	12	1	inner
195.bc	odd	12	1	inner
195.bn	odd	12	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3900.1.dp.a	yes	8
3.b	odd	2	1	CM	3900.1.dp.a	yes	8
5.b	even	2	1	inner	3900.1.dp.a	yes	8
5.c	odd	4	2		3900.1.cu.a	✓	8
13.f	odd	12	1		3900.1.cu.a	✓	8
15.d	odd	2	1	inner	3900.1.dp.a	yes	8
15.e	even	4	2		3900.1.cu.a	✓	8
39.k	even	12	1		3900.1.cu.a	✓	8
65.o	even	12	1	inner	3900.1.dp.a	yes	8
65.s	odd	12	1		3900.1.cu.a	✓	8
65.t	even	12	1	inner	3900.1.dp.a	yes	8
195.bc	odd	12	1	inner	3900.1.dp.a	yes	8
195.bh	even	12	1		3900.1.cu.a	✓	8
195.bn	odd	12	1	inner	3900.1.dp.a	yes	8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
3900.1.cu.a	✓	8	5.c	odd	4	2
3900.1.cu.a	✓	8	13.f	odd	12	1
3900.1.cu.a	✓	8	15.e	even	4	2
3900.1.cu.a	✓	8	39.k	even	12	1
3900.1.cu.a	✓	8	65.s	odd	12	1
3900.1.cu.a	✓	8	195.bh	even	12	1
3900.1.dp.a	yes	8	1.a	even	1	1	trivial
3900.1.dp.a	yes	8	3.b	odd	2	1	CM
3900.1.dp.a	yes	8	5.b	even	2	1	inner
3900.1.dp.a	yes	8	15.d	odd	2	1	inner
3900.1.dp.a	yes	8	65.o	even	12	1	inner
3900.1.dp.a	yes	8	65.t	even	12	1	inner
3900.1.dp.a	yes	8	195.bc	odd	12	1	inner
3900.1.dp.a	yes	8	195.bn	odd	12	1	inner

This newform subspace can be constructed as the kernel of the linear operator $ T_{7}^{8} - 4T_{7}^{6} + 15T_{7}^{4} - 4T_{7}^{2} + 1 $ T7^8 - 4*T7^6 + 15*T7^4 - 4*T7^2 + 1 acting on $S_{1}^{\mathrm{new}}(3900, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{8} $ T^8
$3$	$ T^{8} - T^{4} + 1 $ T^8 - T^4 + 1
$5$	$ T^{8} $ T^8
$7$	$ T^{8} - 4 T^{6} + \cdots + 1 $ T^8 - 4T^6 + 15T^4 - 4*T^2 + 1
$11$	$ T^{8} $ T^8
$13$	$ T^{8} - T^{4} + 1 $ T^8 - T^4 + 1
$17$	$ T^{8} $ T^8
$19$	$ (T^{4} - 2 T^{3} + 2 T^{2} + \cdots + 4)^{2} $ (T^4 - 2T^3 + 2T^2 - 4*T + 4)^2
$23$	$ T^{8} $ T^8
$29$	$ T^{8} $ T^8
$31$	$ (T^{4} - 2 T^{3} + 2 T^{2} + \cdots + 1)^{2} $ (T^4 - 2T^3 + 2T^2 + 2*T + 1)^2
$37$	$ (T^{4} - 2 T^{2} + 4)^{2} $ (T^4 - 2*T^2 + 4)^2
$41$	$ T^{8} $ T^8
$43$	$ T^{8} - T^{4} + 1 $ T^8 - T^4 + 1
$47$	$ T^{8} $ T^8
$53$	$ T^{8} $ T^8
$59$	$ T^{8} $ T^8
$61$	$ (T^{4} + 3 T^{2} + 9)^{2} $ (T^4 + 3*T^2 + 9)^2
$67$	$ T^{8} + 4 T^{6} + \cdots + 1 $ T^8 + 4T^6 + 15T^4 + 4*T^2 + 1
$71$	$ T^{8} $ T^8
$73$	$ (T^{4} - 4 T^{2} + 1)^{2} $ (T^4 - 4*T^2 + 1)^2
$79$	$ (T^{2} + 3)^{4} $ (T^2 + 3)^4
$83$	$ T^{8} $ T^8
$89$	$ T^{8} $ T^8
$97$	$ T^{8} + 4 T^{6} + \cdots + 1 $ T^8 + 4T^6 + 15T^4 + 4*T^2 + 1
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Overview	Random
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Classical	Maass
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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 3900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3900.dp (of order \(12\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\( q + \zeta_{24}^{5} q^{3} + ( - \zeta_{24}^{5} + \zeta_{24}^{3}) q^{7} + \zeta_{24}^{10} q^{9}+O(q^{10}) \) q + z^5 * q^3 + (-z^5 + z^3) * q^7 + z^10 * q^9	\( q + \zeta_{24}^{5} q^{3} + ( - \zeta_{24}^{5} + \zeta_{24}^{3}) q^{7} + \zeta_{24}^{10} q^{9} + \zeta_{24} q^{13} + (\zeta_{24}^{10} + \zeta_{24}^{4}) q^{19} + ( - \zeta_{24}^{10} + \zeta_{24}^{8}) q^{21} - \zeta_{24}^{3} q^{27} + (\zeta_{24}^{4} - \zeta_{24}^{2}) q^{31} + ( - \zeta_{24}^{11} + \zeta_{24}^{5}) q^{37} + \zeta_{24}^{6} q^{39} - \zeta_{24}^{7} q^{43} + (\zeta_{24}^{10} + \cdots + \zeta_{24}^{6}) q^{49} + \cdots + (\zeta_{24}^{5} + \zeta_{24}^{3}) q^{97} +O(q^{100}) \) q + z^5 * q^3 + (-z^5 + z^3) * q^7 + z^10 * q^9 + z * q^13 + (z^10 + z^4) * q^19 + (-z^10 + z^8) * q^21 - z^3 * q^27 + (z^4 - z^2) * q^31 + (-z^11 + z^5) * q^37 + z^6 * q^39 - z^7 * q^43 + (z^10 - z^8 + z^6) * q^49 + (z^9 - z^3) * q^57 + (z^6 + z^2) * q^61 + (z^3 - z) * q^63 + (-z^9 - z^7) * q^67 + (z^7 - z^5) * q^73 + (z^8 + z^4) * q^79 - z^8 * q^81 + (-z^6 + z^4) * q^91 + (z^9 - z^7) * q^93 + (z^5 + z^3) * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 4 q^{19} - 4 q^{21} + 4 q^{31} + 4 q^{49} + 4 q^{81} + 4 q^{91}+O(q^{100}) \) 8 * q + 4 * q^19 - 4 * q^21 + 4 * q^31 + 4 * q^49 + 4 * q^81 + 4 * q^91

Introduction

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Modular forms

Varieties

Fields

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	3900.1.dp.a

Level	$3900$
Weight	$1$
Character orbit	3900.dp
Analytic conductor	$1.946$
Analytic rank	$0$
Dimension	$8$
Projective image	$D_{12}$
CM discriminant	-3
Inner twists	$8$

Self dual:	no
Analytic conductor:	\(1.94635354927\)
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:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{12}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{12} - \cdots)\)

\(n\)	\(301\)	\(1301\)	\(1951\)	\(3277\)
\(\chi(n)\)	\(\zeta_{24}^{10}\)	\(-1\)	\(1\)	\(\zeta_{24}^{6}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 293.2
Significant digits:
Format:

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by \(\Q(\sqrt{-3}) \)
5.b	even	2	1	inner
15.d	odd	2	1	inner
65.o	even	12	1	inner
65.t	even	12	1	inner
195.bc	odd	12	1	inner
195.bn	odd	12	1	inner

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