LMFDB - Newform orbit 3069.1.cd.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3069,1,Mod(433,3069)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3069, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([0, 2, 5]))

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

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

chi := DirichletCharacter("3069.433");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$N$	$=$	$3069 = 3^{2} \cdot 11 \cdot 31$
Weight:	$k$	$=$	$1$
Character orbit:	$[\chi]$	$=$	3069.cd (of order $10$ , 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:	$1.53163052377$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$2$ over $\Q(\zeta_{10})$
Coefficient field:	$\Q(\zeta_{20})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{8} - x^{6} + x^{4} - x^{2} + 1$ x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$1$
Twist minimal:	yes
Projective image:	$A_{5}$
Projective field:	Galois closure of 5.1.126630009.1

$q$ -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.

$f(q)$	$=$	$q + (\zeta_{20}^{2} - 1) q^{2} + (\zeta_{20}^{4} - \zeta_{20}^{2} + 1) q^{4} + \zeta_{20}^{4} q^{5} + \zeta_{20}^{2} q^{7} + (\zeta_{20}^{6} - \zeta_{20}^{4} + \cdots - 1) q^{8} + (\zeta_{20}^{6} - \zeta_{20}^{4}) q^{10}+ \cdots - \zeta_{20}^{6} q^{97} +O(q^{100})$ q + (z^2 - 1) * q^2 + (z^4 - z^2 + 1) * q^4 + z^4 * q^5 + z^2 * q^7 + (z^6 - z^4 + z^2 - 1) * q^8 + (z^6 - z^4) * q^10 - z^7 * q^11 - z * q^13 + (z^4 - z^2) * q^14 + (z^8 - z^6 + z^4 - z^2 + 1) * q^16 + (z^5 - z^3) * q^17 + (z^8 - z^6 + z^4) * q^20 + (-z^9 + z^7) * q^22 + (z^9 + z) * q^23 + (-z^3 + z) * q^26 + (z^6 - z^4 + z^2) * q^28 + (z^9 + z^5) * q^29 + z * q^31 + (-z^8 + z^6 - z^4 + z^2 - 2) * q^32 + (z^7 - 2z^5 + z^3) q^34 + z^6 * q^35 - z^7 * q^37 + (-z^8 + z^6 - z^4 - 1) * q^40 + (z^7 + z^3) * q^43 + (z^9 - z^7 + z) * q^44 + (-z^9 + z^3 - 2z) q^46 + (-z^4 + z^2) * q^47 + (-z^5 + z^3 - z) * q^52 + z * q^55 + (z^8 - z^6 + z^4 - z^2) * q^56 + (-z^9 + z^7 - z^5 - z) * q^58 - z^9 * q^61 + (z^3 - z) * q^62 + (z^8 - z^6 + z^4 - 2z^2 + 2) q^64 - z^5 * q^65 - q^67 + (z^9 - 2z^7 + 2z^5 - z^3) * q^68 + (z^8 - z^6) * q^70 - z^4 * q^71 + z^7 * q^73 + (-z^9 + z^7) * q^74 - z^9 * q^77 - z * q^79 + (z^8 - z^6 + z^4 - z^2 + 1) * q^80 - z^9 * q^83 + (z^9 - z^7) * q^85 + (z^9 - z^7 + z^5 - z^3) * q^86 + (-z^9 + z^7 + z^3 - z) * q^88 - z^5 * q^89 - z^3 * q^91 + (z^9 + z^5 - 2z^3 + 2z) * q^92 + (-z^6 + 2z^4 - z^2) q^94 - z^6 * q^97
$\operatorname{Tr}(f)(q)$	$=$	$8 q - 6 q^{2} + 4 q^{4} - 2 q^{5} + 2 q^{7} - 2 q^{8} + 4 q^{10} - 4 q^{14} - 6 q^{20} + 6 q^{28} - 8 q^{32} + 2 q^{35} - 2 q^{40} + 4 q^{47} - 8 q^{56} + 6 q^{64} - 8 q^{67} - 4 q^{70} + 2 q^{71} - 8 q^{94}+ \cdots - 2 q^{97}+O(q^{100})$ 8 * q - 6 * q^2 + 4 * q^4 - 2 * q^5 + 2 * q^7 - 2 * q^8 + 4 * q^10 - 4 * q^14 - 6 * q^20 + 6 * q^28 - 8 * q^32 + 2 * q^35 - 2 * q^40 + 4 * q^47 - 8 * q^56 + 6 * q^64 - 8 * q^67 - 4 * q^70 + 2 * q^71 - 8 * q^94 - 2 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3069\mathbb{Z}\right)^\times$ .

$n$	$838$	$2080$	$2729$
$\chi(n)$	$-\zeta_{20}^{2}$	$-1$	$1$

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}

433.1

0.587785	−	0.809017i
−0.587785	+	0.809017i
−0.951057	+	0.309017i
0.951057	−	0.309017i
−0.951057	−	0.309017i
0.951057	+	0.309017i
0.587785	+	0.809017i
−0.587785	−	0.809017i

−1.30902

−

0.951057i

0.500000

1.53884i

−0.809017

0.587785i

−0.309017

−

0.951057i

0.309017

−

0.951057i

1.61803

433.2

−1.30902

−

0.951057i

0.500000

1.53884i

−0.809017

0.587785i

−0.309017

−

0.951057i

0.309017

−

0.951057i

1.61803

1270.1

−0.190983

−

0.587785i

0.500000

−

0.363271i

0.309017

−

0.951057i

0.809017

−

0.587785i

−0.809017

−

0.587785i

−0.618034

1270.2

−0.190983

−

0.587785i

0.500000

−

0.363271i

0.309017

−

0.951057i

0.809017

−

0.587785i

−0.809017

−

0.587785i

−0.618034

1549.1

−0.190983

0.587785i

0.500000

0.363271i

0.309017

0.951057i

0.809017

0.587785i

−0.809017

0.587785i

−0.618034

1549.2

−0.190983

0.587785i

0.500000

0.363271i

0.309017

0.951057i

0.809017

0.587785i

−0.809017

0.587785i

−0.618034

2665.1

−1.30902

0.951057i

0.500000

−

1.53884i

−0.809017

−

0.587785i

−0.309017

0.951057i

0.309017

0.951057i

1.61803

2665.2

−1.30902

0.951057i

0.500000

−

1.53884i

−0.809017

−

0.587785i

−0.309017

0.951057i

0.309017

0.951057i

1.61803

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
11.c	even	5	1	inner
31.b	odd	2	1	inner
341.t	odd	10	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3069.1.cd.a	✓	8
3.b	odd	2	1		3069.1.cd.b	yes	8
11.c	even	5	1	inner	3069.1.cd.a	✓	8
31.b	odd	2	1	inner	3069.1.cd.a	✓	8
33.h	odd	10	1		3069.1.cd.b	yes	8
93.c	even	2	1		3069.1.cd.b	yes	8
341.t	odd	10	1	inner	3069.1.cd.a	✓	8
1023.cg	even	10	1		3069.1.cd.b	yes	8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
3069.1.cd.a	✓	8	1.a	even	1	1	trivial
3069.1.cd.a	✓	8	11.c	even	5	1	inner
3069.1.cd.a	✓	8	31.b	odd	2	1	inner
3069.1.cd.a	✓	8	341.t	odd	10	1	inner
3069.1.cd.b	yes	8	3.b	odd	2	1
3069.1.cd.b	yes	8	33.h	odd	10	1
3069.1.cd.b	yes	8	93.c	even	2	1
3069.1.cd.b	yes	8	1023.cg	even	10	1

This newform subspace can be constructed as the kernel of the linear operator $T_{2}^{4} + 3T_{2}^{3} + 4T_{2}^{2} + 2T_{2} + 1$ T2^4 + 3*T2^3 + 4*T2^2 + 2*T2 + 1 acting on $S_{1}^{\mathrm{new}}(3069, [\chi])$ .

Hecke characteristic polynomials

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

$n$ :		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 433.2
Significant digits:
Format:

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

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	3069.1.cd.a

Level	$3069$
Weight	$1$
Character orbit	3069.cd
Analytic conductor	$1.532$
Analytic rank	$0$
Dimension	$8$
Projective image	$A_{5}$
CM/RM	no
Inner twists	$4$