LMFDB - Newform orbit 1332.1.cy.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1332,1,Mod(559,1332)] 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("1332.559"); S:= CuspForms(chi, 1); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1332, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([9, 0, 1])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]

Level:	$N$	$=$	$1332 = 2^{2} \cdot 3^{2} \cdot 37$
Weight:	$k$	$=$	$1$
Character orbit:	$[\chi]$	$=$	1332.cy (of order $18$ , degree $6$ , 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:	$0.664754596827$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$2$ over $\Q(\zeta_{18})$
Coefficient field:	$\Q(\zeta_{36})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{12} - x^{6} + 1$ x^12 - x^6 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$1$
Twist minimal:	yes
Projective image:	$D_{18}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{18} - \cdots)$

$q$ -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.

$f(q)$	$=$	$q + \zeta_{36}^{13} q^{2} - \zeta_{36}^{8} q^{4} + ( - \zeta_{36}^{15} - \zeta_{36}^{7}) q^{5} + \zeta_{36}^{3} q^{8} + (\zeta_{36}^{10} + \zeta_{36}^{2}) q^{10} + ( - \zeta_{36}^{14} + \zeta_{36}^{2}) q^{13} + \cdots + \zeta_{36}^{17} q^{98} +O(q^{100})$ q + z^13 * q^2 - z^8 * q^4 + (-z^15 - z^7) * q^5 + z^3 * q^8 + (z^10 + z^2) * q^10 + (-z^14 + z^2) * q^13 + z^16 * q^16 + (z^17 - z^3) * q^17 + (z^15 - z^5) * q^20 + (z^14 - z^12 - z^4) * q^25 + (z^15 + z^9) * q^26 + (-z^13 + z^11) * q^29 - z^11 * q^32 + (-z^16 - z^12) * q^34 - z^8 * q^37 + (-z^10 + 1) * q^40 + (-z^15 - z) * q^41 + z^4 * q^49 + (-z^17 - z^9 + z^7) * q^50 + (-z^10 - z^4) * q^52 + (-z^13 + z) * q^53 + (z^8 - z^6) * q^58 + (z^16 - 1) * q^61 + z^6 * q^64 + (-z^17 - z^11 - z^9 - z^3) * q^65 + (z^11 + z^7) * q^68 - q^73 + z^3 * q^74 + (z^13 + z^5) * q^80 + (-z^14 + z^10) * q^82 + (z^14 + z^10 + z^6 - 1) * q^85 + (z^9 + z^5) * q^89 + (-z^16 - z^14) * q^97 + z^17 * q^98
$\operatorname{Tr}(f)(q)$	$=$	$12 q + 6 q^{25} + 6 q^{34} + 12 q^{40} - 6 q^{58} - 12 q^{61} + 6 q^{64} - 12 q^{73} - 6 q^{85}+O(q^{100})$ 12 * q + 6 * q^25 + 6 * q^34 + 12 * q^40 - 6 * q^58 - 12 * q^61 + 6 * q^64 - 12 * q^73 - 6 * q^85

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1332\mathbb{Z}\right)^\times$ .

$n$	$667$	$1037$	$1297$
$\chi(n)$	$-1$	$1$	$-\zeta_{36}^{8}$

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}

559.1

0.342020	+	0.939693i
−0.342020	−	0.939693i
0.984808	+	0.173648i
−0.984808	−	0.173648i
−0.642788	+	0.766044i
0.642788	−	0.766044i
0.342020	−	0.939693i
−0.342020	+	0.939693i
0.984808	−	0.173648i
−0.984808	+	0.173648i
−0.642788	−	0.766044i
0.642788	+	0.766044i

−0.984808

−

0.173648i

0.939693

0.342020i

−0.223238

−

0.266044i

−0.866025

−

0.500000i

0.173648

0.300767i

559.2

0.984808

0.173648i

0.939693

0.342020i

0.223238

0.266044i

0.866025

0.500000i

0.173648

0.300767i

595.1

−0.642788

0.766044i

−0.173648

−

0.984808i

0.524005

−

1.43969i

0.866025

0.500000i

0.766044

1.32683i

595.2

0.642788

−

0.766044i

−0.173648

−

0.984808i

−0.524005

1.43969i

−0.866025

−

0.500000i

0.766044

1.32683i

955.1

−0.342020

−

0.939693i

−0.766044

0.642788i

1.85083

−

0.326352i

0.866025

0.500000i

−0.939693

−

1.62760i

955.2

0.342020

0.939693i

−0.766044

0.642788i

−1.85083

0.326352i

−0.866025

−

0.500000i

−0.939693

−

1.62760i

1027.1

−0.984808

0.173648i

0.939693

−

0.342020i

−0.223238

0.266044i

−0.866025

0.500000i

0.173648

−

0.300767i

1027.2

0.984808

−

0.173648i

0.939693

−

0.342020i

0.223238

−

0.266044i

0.866025

−

0.500000i

0.173648

−

0.300767i

1135.1

−0.642788

−

0.766044i

−0.173648

0.984808i

0.524005

1.43969i

0.866025

−

0.500000i

0.766044

−

1.32683i

1135.2

0.642788

0.766044i

−0.173648

0.984808i

−0.524005

−

1.43969i

−0.866025

0.500000i

0.766044

−

1.32683i

1279.1

−0.342020

0.939693i

−0.766044

−

0.642788i

1.85083

0.326352i

0.866025

−

0.500000i

−0.939693

1.62760i

1279.2

0.342020

−

0.939693i

−0.766044

−

0.642788i

−1.85083

−

0.326352i

−0.866025

0.500000i

−0.939693

1.62760i

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
37.h	even	18	1	inner
111.n	odd	18	1	inner
148.o	odd	18	1	inner
444.ba	even	18	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1332.1.cy.a	✓	12
3.b	odd	2	1	inner	1332.1.cy.a	✓	12
4.b	odd	2	1	CM	1332.1.cy.a	✓	12
12.b	even	2	1	inner	1332.1.cy.a	✓	12
37.h	even	18	1	inner	1332.1.cy.a	✓	12
111.n	odd	18	1	inner	1332.1.cy.a	✓	12
148.o	odd	18	1	inner	1332.1.cy.a	✓	12
444.ba	even	18	1	inner	1332.1.cy.a	✓	12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1332.1.cy.a	✓	12	1.a	even	1	1	trivial
1332.1.cy.a	✓	12	3.b	odd	2	1	inner
1332.1.cy.a	✓	12	4.b	odd	2	1	CM
1332.1.cy.a	✓	12	12.b	even	2	1	inner
1332.1.cy.a	✓	12	37.h	even	18	1	inner
1332.1.cy.a	✓	12	111.n	odd	18	1	inner
1332.1.cy.a	✓	12	148.o	odd	18	1	inner
1332.1.cy.a	✓	12	444.ba	even	18	1	inner

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(1332, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{12} - T^{6} + 1$ T^12 - T^6 + 1
$3$	$T^{12}$ T^12
$5$	$T^{12} - 3 T^{10} + \cdots + 1$ T^12 - 3T^10 - 6T^8 + 8T^6 + 69T^4 + 3*T^2 + 1
$7$	$T^{12}$ T^12
$11$	$T^{12}$ T^12
$13$	$(T^{6} - 9 T^{3} + 27)^{2}$ (T^6 - 9*T^3 + 27)^2
$17$	$T^{12} - 3 T^{10} + \cdots + 1$ T^12 - 3T^10 - 6T^8 + 8T^6 + 69T^4 + 3*T^2 + 1
$19$	$T^{12}$ T^12
$23$	$T^{12}$ T^12
$29$	$T^{12} - 6 T^{10} + \cdots + 1$ T^12 - 6T^10 + 27T^8 - 52T^6 + 75T^4 - 9*T^2 + 1
$31$	$T^{12}$ T^12
$37$	$(T^{6} - T^{3} + 1)^{2}$ (T^6 - T^3 + 1)^2
$41$	$T^{12} - 3 T^{10} + \cdots + 9$ T^12 - 3T^10 + 18T^8 - 24T^6 + 9T^4 + 27*T^2 + 9
$43$	$T^{12}$ T^12
$47$	$T^{12}$ T^12
$53$	$T^{12} + 27T^{6} + 729$ T^12 + 27*T^6 + 729
$59$	$T^{12}$ T^12
$61$	$(T^{6} + 6 T^{5} + 15 T^{4} + \cdots + 3)^{2}$ (T^6 + 6T^5 + 15T^4 + 21T^3 + 18T^2 + 9*T + 3)^2
$67$	$T^{12}$ T^12
$71$	$T^{12}$ T^12
$73$	$(T + 1)^{12}$ (T + 1)^12
$79$	$T^{12}$ T^12
$83$	$T^{12}$ T^12
$89$	$T^{12} + 6 T^{10} + \cdots + 1$ T^12 + 6T^10 + 21T^8 + 35T^6 + 60T^4 - 15*T^2 + 1
$97$	$(T^{6} - 3 T^{4} + 9 T^{2} + \cdots + 3)^{2}$ (T^6 - 3T^4 + 9T^2 + 9*T + 3)^2
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Elliptic curves over $\Q$
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Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

$n$ :		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 559.2
Significant digits:
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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	1332.1.cy.a

Level	$1332$
Weight	$1$
Character orbit	1332.cy
Analytic conductor	$0.665$
Analytic rank	$0$
Dimension	$12$
Projective image	$D_{18}$
CM discriminant	-4
Inner twists	$8$