LMFDB - Newform orbit 2916.1.j.c

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2916,1,Mod(163,2916)] 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("2916.163"); S:= CuspForms(chi, 1); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2916, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([9, 16])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]

Level:	$ N $	$=$	$ 2916 = 2^{2} \cdot 3^{6} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2916.j (of order $18$, degree $6$, not minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,0,0,-3,0,0,-3,0,0,0,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	no
Analytic conductor:	$1.45527357684$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\Q(\zeta_{18})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{3} + 1 $ x^6 - x^3 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{9}$
Projective field:	Galois closure of 9.1.8033551259904.2

$q$-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_{18}^{4} q^{2} + \zeta_{18}^{8} q^{4} + ( - \zeta_{18}^{3} - \zeta_{18}) q^{5} - \zeta_{18}^{3} q^{8} + ( - \zeta_{18}^{7} - \zeta_{18}^{5}) q^{10} + ( - \zeta_{18} + 1) q^{13} - \zeta_{18}^{7} q^{16} + \cdots + \zeta_{18}^{6} q^{98} +O(q^{100}) $ q + z^4 * q^2 + z^8 * q^4 + (-z^3 - z) * q^5 - z^3 * q^8 + (-z^7 - z^5) * q^10 + (-z + 1) * q^13 - z^7 * q^16 + (z^8 + z^4) * q^17 + (z^2 + 1) * q^20 + (z^6 + z^4 + z^2) * q^25 + (-z^5 + z^4) * q^26 + (z^8 + 1) * q^29 + z^2 * q^32 + (z^8 - z^3) * q^34 + (z^8 + z^4) * q^37 + (z^6 + z^4) * q^40 + z^5 * q^41 + z^2 * q^49 + (z^8 + z^6 - z) * q^50 + (z^8 + 1) * q^52 - q^53 + (z^4 - z^3) * q^58 + (z^6 - z^5) * q^61 + z^6 * q^64 + (z^4 - z^3 + z^2 - z) * q^65 + (-z^7 - z^3) * q^68 + (z^4 + z^2) * q^73 + (z^8 - z^3) * q^74 + (z^8 - z) * q^80 - q^82 + (-z^7 - z^5 + z^2 + 1) * q^85 + (-z^5 - z) * q^89 + z^7 * q^97 + z^6 * q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 3 q^{5} - 3 q^{8} + 6 q^{13} + 6 q^{20} - 3 q^{25} + 6 q^{29} - 3 q^{34} - 3 q^{40} - 3 q^{50} + 6 q^{52} - 6 q^{53} - 3 q^{58} - 3 q^{61} - 3 q^{64} - 3 q^{65} - 3 q^{68} - 3 q^{74} - 6 q^{82} + 6 q^{85}+ \cdots - 3 q^{98}+O(q^{100}) $ 6 * q - 3 * q^5 - 3 * q^8 + 6 * q^13 + 6 * q^20 - 3 * q^25 + 6 * q^29 - 3 * q^34 - 3 * q^40 - 3 * q^50 + 6 * q^52 - 6 * q^53 - 3 * q^58 - 3 * q^61 - 3 * q^64 - 3 * q^65 - 3 * q^68 - 3 * q^74 - 6 * q^82 + 6 * q^85 - 3 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2916\mathbb{Z}\right)^\times$.

$n$	$1459$	$2189$
$\chi(n)$	$-1$	$\zeta_{18}^{8}$

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} $

163.1

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

−0.939693

0.342020i

0.766044

−

0.642788i

0.266044

1.50881i

−0.500000

0.866025i

−0.766044

−

1.32683i

811.1

0.173648

0.984808i

−0.939693

0.342020i

−1.43969

−

1.20805i

−0.500000

−

0.866025i

0.939693

−

1.62760i

1135.1

0.766044

0.642788i

0.173648

0.984808i

−0.326352

−

0.118782i

−0.500000

0.866025i

−0.173648

−

0.300767i

1783.1

0.766044

−

0.642788i

0.173648

−

0.984808i

−0.326352

0.118782i

−0.500000

−

0.866025i

−0.173648

0.300767i

2107.1

0.173648

−

0.984808i

−0.939693

−

0.342020i

−1.43969

1.20805i

−0.500000

0.866025i

0.939693

1.62760i

2755.1

−0.939693

−

0.342020i

0.766044

0.642788i

0.266044

−

1.50881i

−0.500000

−

0.866025i

−0.766044

1.32683i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	CM by $\Q(\sqrt{-1}) $
27.e	even	9	1	inner
108.j	odd	18	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2916.1.j.c		6
3.b	odd	2	1		2916.1.j.g		6
4.b	odd	2	1	CM	2916.1.j.c		6
9.c	even	3	1		2916.1.j.b		6
9.c	even	3	1		2916.1.j.h		6
9.d	odd	6	1		2916.1.j.a		6
9.d	odd	6	1		2916.1.j.f		6
12.b	even	2	1		2916.1.j.g		6
27.e	even	9	1		2916.1.d.b	yes	3
27.e	even	9	2		2916.1.f.a		6
27.e	even	9	1		2916.1.j.b		6
27.e	even	9	1	inner	2916.1.j.c		6
27.e	even	9	1		2916.1.j.h		6
27.f	odd	18	1		2916.1.d.a	✓	3
27.f	odd	18	2		2916.1.f.b		6
27.f	odd	18	1		2916.1.j.a		6
27.f	odd	18	1		2916.1.j.f		6
27.f	odd	18	1		2916.1.j.g		6
36.f	odd	6	1		2916.1.j.b		6
36.f	odd	6	1		2916.1.j.h		6
36.h	even	6	1		2916.1.j.a		6
36.h	even	6	1		2916.1.j.f		6
108.j	odd	18	1		2916.1.d.b	yes	3
108.j	odd	18	2		2916.1.f.a		6
108.j	odd	18	1		2916.1.j.b		6
108.j	odd	18	1	inner	2916.1.j.c		6
108.j	odd	18	1		2916.1.j.h		6
108.l	even	18	1		2916.1.d.a	✓	3
108.l	even	18	2		2916.1.f.b		6
108.l	even	18	1		2916.1.j.a		6
108.l	even	18	1		2916.1.j.f		6
108.l	even	18	1		2916.1.j.g		6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
2916.1.d.a	✓	3	27.f	odd	18	1
2916.1.d.a	✓	3	108.l	even	18	1
2916.1.d.b	yes	3	27.e	even	9	1
2916.1.d.b	yes	3	108.j	odd	18	1
2916.1.f.a		6	27.e	even	9	2
2916.1.f.a		6	108.j	odd	18	2
2916.1.f.b		6	27.f	odd	18	2
2916.1.f.b		6	108.l	even	18	2
2916.1.j.a		6	9.d	odd	6	1
2916.1.j.a		6	27.f	odd	18	1
2916.1.j.a		6	36.h	even	6	1
2916.1.j.a		6	108.l	even	18	1
2916.1.j.b		6	9.c	even	3	1
2916.1.j.b		6	27.e	even	9	1
2916.1.j.b		6	36.f	odd	6	1
2916.1.j.b		6	108.j	odd	18	1
2916.1.j.c		6	1.a	even	1	1	trivial
2916.1.j.c		6	4.b	odd	2	1	CM
2916.1.j.c		6	27.e	even	9	1	inner
2916.1.j.c		6	108.j	odd	18	1	inner
2916.1.j.f		6	9.d	odd	6	1
2916.1.j.f		6	27.f	odd	18	1
2916.1.j.f		6	36.h	even	6	1
2916.1.j.f		6	108.l	even	18	1
2916.1.j.g		6	3.b	odd	2	1
2916.1.j.g		6	12.b	even	2	1
2916.1.j.g		6	27.f	odd	18	1
2916.1.j.g		6	108.l	even	18	1
2916.1.j.h		6	9.c	even	3	1
2916.1.j.h		6	27.e	even	9	1
2916.1.j.h		6	36.f	odd	6	1
2916.1.j.h		6	108.j	odd	18	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{6} + 3T_{5}^{5} + 6T_{5}^{4} + 8T_{5}^{3} + 12T_{5}^{2} + 6T_{5} + 1 $ T5^6 + 3*T5^5 + 6*T5^4 + 8*T5^3 + 12*T5^2 + 6*T5 + 1 acting on $S_{1}^{\mathrm{new}}(2916, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{6} + T^{3} + 1 $ T^6 + T^3 + 1
$3$	$ T^{6} $ T^6
$5$	$ T^{6} + 3 T^{5} + \cdots + 1 $ T^6 + 3T^5 + 6T^4 + 8T^3 + 12T^2 + 6*T + 1
$7$	$ T^{6} $ T^6
$11$	$ T^{6} $ T^6
$13$	$ T^{6} - 6 T^{5} + \cdots + 1 $ T^6 - 6T^5 + 15T^4 - 19T^3 + 12T^2 - 3*T + 1
$17$	$ T^{6} + 3 T^{4} + \cdots + 1 $ T^6 + 3T^4 + 2T^3 + 9T^2 + 3T + 1
$19$	$ T^{6} $ T^6
$23$	$ T^{6} $ T^6
$29$	$ T^{6} - 6 T^{5} + \cdots + 1 $ T^6 - 6T^5 + 15T^4 - 19T^3 + 12T^2 - 3*T + 1
$31$	$ T^{6} $ T^6
$37$	$ T^{6} + 3 T^{4} + \cdots + 1 $ T^6 + 3T^4 + 2T^3 + 9T^2 + 3T + 1
$41$	$ T^{6} - T^{3} + 1 $ T^6 - T^3 + 1
$43$	$ T^{6} $ T^6
$47$	$ T^{6} $ T^6
$53$	$ (T + 1)^{6} $ (T + 1)^6
$59$	$ T^{6} $ T^6
$61$	$ T^{6} + 3 T^{5} + \cdots + 1 $ T^6 + 3T^5 + 6T^4 + 8T^3 + 12T^2 + 6*T + 1
$67$	$ T^{6} $ T^6
$71$	$ T^{6} $ T^6
$73$	$ T^{6} + 3 T^{4} + \cdots + 1 $ T^6 + 3T^4 + 2T^3 + 9T^2 + 3T + 1
$79$	$ T^{6} $ T^6
$83$	$ T^{6} $ T^6
$89$	$ T^{6} + 3 T^{4} + \cdots + 1 $ T^6 + 3T^4 + 2T^3 + 9T^2 + 3T + 1
$97$	$ T^{6} - T^{3} + 1 $ T^6 - T^3 + 1
show more
show less

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$ over number fields
Genus 2 curves
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2916 = 2^{2} \cdot 3^{6} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2916.j (of order \(18\), degree \(6\), not minimal)

\(f(q)\)	\(=\)	\( q + \zeta_{18}^{4} q^{2} + \zeta_{18}^{8} q^{4} + ( - \zeta_{18}^{3} - \zeta_{18}) q^{5} - \zeta_{18}^{3} q^{8} + ( - \zeta_{18}^{7} - \zeta_{18}^{5}) q^{10} + ( - \zeta_{18} + 1) q^{13} - \zeta_{18}^{7} q^{16} + \cdots + \zeta_{18}^{6} q^{98} +O(q^{100}) \) q + z^4 * q^2 + z^8 * q^4 + (-z^3 - z) * q^5 - z^3 * q^8 + (-z^7 - z^5) * q^10 + (-z + 1) * q^13 - z^7 * q^16 + (z^8 + z^4) * q^17 + (z^2 + 1) * q^20 + (z^6 + z^4 + z^2) * q^25 + (-z^5 + z^4) * q^26 + (z^8 + 1) * q^29 + z^2 * q^32 + (z^8 - z^3) * q^34 + (z^8 + z^4) * q^37 + (z^6 + z^4) * q^40 + z^5 * q^41 + z^2 * q^49 + (z^8 + z^6 - z) * q^50 + (z^8 + 1) * q^52 - q^53 + (z^4 - z^3) * q^58 + (z^6 - z^5) * q^61 + z^6 * q^64 + (z^4 - z^3 + z^2 - z) * q^65 + (-z^7 - z^3) * q^68 + (z^4 + z^2) * q^73 + (z^8 - z^3) * q^74 + (z^8 - z) * q^80 - q^82 + (-z^7 - z^5 + z^2 + 1) * q^85 + (-z^5 - z) * q^89 + z^7 * q^97 + z^6 * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 3 q^{5} - 3 q^{8} + 6 q^{13} + 6 q^{20} - 3 q^{25} + 6 q^{29} - 3 q^{34} - 3 q^{40} - 3 q^{50} + 6 q^{52} - 6 q^{53} - 3 q^{58} - 3 q^{61} - 3 q^{64} - 3 q^{65} - 3 q^{68} - 3 q^{74} - 6 q^{82} + 6 q^{85}+ \cdots - 3 q^{98}+O(q^{100}) \) 6 * q - 3 * q^5 - 3 * q^8 + 6 * q^13 + 6 * q^20 - 3 * q^25 + 6 * q^29 - 3 * q^34 - 3 * q^40 - 3 * q^50 + 6 * q^52 - 6 * q^53 - 3 * q^58 - 3 * q^61 - 3 * q^64 - 3 * q^65 - 3 * q^68 - 3 * q^74 - 6 * q^82 + 6 * q^85 - 3 * q^98

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	2916.1.j.c

Level	$2916$
Weight	$1$
Character orbit	2916.j
Analytic conductor	$1.455$
Analytic rank	$0$
Dimension	$6$
Projective image	$D_{9}$
CM discriminant	-4
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(1.45527357684\)
Analytic rank:	\(0\)
Dimension:	\(6\)
Coefficient field:	\(\Q(\zeta_{18})\)
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{6} - x^{3} + 1 \) x^6 - x^3 + 1
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{9}\)
Projective field:	Galois closure of 9.1.8033551259904.2

\(n\):		e.g. 2-40 or 80-90
Embeddings:		e.g. 1-3 or 163.1
Significant digits:
Format:

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	CM by \(\Q(\sqrt{-1}) \)
27.e	even	9	1	inner
108.j	odd	18	1	inner

$p$	$F_p(T)$
$2$	\( T^{6} + T^{3} + 1 \) T^6 + T^3 + 1
$3$	\( T^{6} \) T^6
$5$	\( T^{6} + 3 T^{5} + \cdots + 1 \) T^6 + 3T^5 + 6T^4 + 8T^3 + 12T^2 + 6*T + 1
$7$	\( T^{6} \) T^6
$11$	\( T^{6} \) T^6
$13$	\( T^{6} - 6 T^{5} + \cdots + 1 \) T^6 - 6T^5 + 15T^4 - 19T^3 + 12T^2 - 3*T + 1
$17$	\( T^{6} + 3 T^{4} + \cdots + 1 \) T^6 + 3T^4 + 2T^3 + 9T^2 + 3T + 1
$19$	\( T^{6} \) T^6
$23$	\( T^{6} \) T^6
$29$	\( T^{6} - 6 T^{5} + \cdots + 1 \) T^6 - 6T^5 + 15T^4 - 19T^3 + 12T^2 - 3*T + 1
$31$	\( T^{6} \) T^6
$37$	\( T^{6} + 3 T^{4} + \cdots + 1 \) T^6 + 3T^4 + 2T^3 + 9T^2 + 3T + 1
$41$	\( T^{6} - T^{3} + 1 \) T^6 - T^3 + 1
$43$	\( T^{6} \) T^6
$47$	\( T^{6} \) T^6
$53$	\( (T + 1)^{6} \) (T + 1)^6
$59$	\( T^{6} \) T^6
$61$	\( T^{6} + 3 T^{5} + \cdots + 1 \) T^6 + 3T^5 + 6T^4 + 8T^3 + 12T^2 + 6*T + 1
$67$	\( T^{6} \) T^6
$71$	\( T^{6} \) T^6
$73$	\( T^{6} + 3 T^{4} + \cdots + 1 \) T^6 + 3T^4 + 2T^3 + 9T^2 + 3T + 1
$79$	\( T^{6} \) T^6
$83$	\( T^{6} \) T^6
$89$	\( T^{6} + 3 T^{4} + \cdots + 1 \) T^6 + 3T^4 + 2T^3 + 9T^2 + 3T + 1
$97$	\( T^{6} - T^{3} + 1 \) T^6 - T^3 + 1
show more
show less

\(n\)	\(1459\)	\(2189\)
\(\chi(n)\)	\(-1\)	\(\zeta_{18}^{8}\)