LMFDB - Newform orbit 1944.1.r.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1944,1,Mod(379,1944)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1944, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([9, 9, 4]))

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

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

chi := DirichletCharacter("1944.379");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1944 = 2^{3} \cdot 3^{5} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1944.r (of order $18$, degree $6$, not 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:	$0.970182384559$
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:	no (minimal twist has level 216)
Projective image:	$D_{9}$
Projective field:	Galois closure of 9.1.128536820158464.7

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1944\mathbb{Z}\right)^\times$.

$n$	$487$	$973$	$1217$
$\chi(n)$	$-1$	$-1$	$\zeta_{18}^{8}$

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} $

379.1

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

−0.766044

−

0.642788i

0.173648

0.984808i

0.500000

−

0.866025i

595.1

−0.766044

0.642788i

0.173648

−

0.984808i

0.500000

0.866025i

1027.1

0.939693

−

0.342020i

0.766044

−

0.642788i

0.500000

−

0.866025i

1243.1

−0.173648

−

0.984808i

−0.939693

0.342020i

0.500000

0.866025i

1675.1

−0.173648

0.984808i

−0.939693

−

0.342020i

0.500000

−

0.866025i

1891.1

0.939693

0.342020i

0.766044

0.642788i

0.500000

0.866025i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.d	odd	2	1	CM by $\Q(\sqrt{-2}) $
27.e	even	9	1	inner
216.r	odd	18	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1944.1.r.c		6
3.b	odd	2	1		1944.1.r.b		6
8.d	odd	2	1	CM	1944.1.r.c		6
9.c	even	3	1		648.1.r.a		6
9.c	even	3	1		1944.1.r.d		6
9.d	odd	6	1		216.1.r.a	✓	6
9.d	odd	6	1		1944.1.r.a		6
24.f	even	2	1		1944.1.r.b		6
27.e	even	9	1		648.1.r.a		6
27.e	even	9	1	inner	1944.1.r.c		6
27.e	even	9	1		1944.1.r.d		6
27.f	odd	18	1		216.1.r.a	✓	6
27.f	odd	18	1		1944.1.r.a		6
27.f	odd	18	1		1944.1.r.b		6
36.f	odd	6	1		2592.1.bd.a		6
36.h	even	6	1		864.1.bd.a		6
72.j	odd	6	1		864.1.bd.a		6
72.l	even	6	1		216.1.r.a	✓	6
72.l	even	6	1		1944.1.r.a		6
72.n	even	6	1		2592.1.bd.a		6
72.p	odd	6	1		648.1.r.a		6
72.p	odd	6	1		1944.1.r.d		6
108.j	odd	18	1		2592.1.bd.a		6
108.l	even	18	1		864.1.bd.a		6
216.r	odd	18	1		648.1.r.a		6
216.r	odd	18	1	inner	1944.1.r.c		6
216.r	odd	18	1		1944.1.r.d		6
216.t	even	18	1		2592.1.bd.a		6
216.v	even	18	1		216.1.r.a	✓	6
216.v	even	18	1		1944.1.r.a		6
216.v	even	18	1		1944.1.r.b		6
216.x	odd	18	1		864.1.bd.a		6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
216.1.r.a	✓	6	9.d	odd	6	1
216.1.r.a	✓	6	27.f	odd	18	1
216.1.r.a	✓	6	72.l	even	6	1
216.1.r.a	✓	6	216.v	even	18	1
648.1.r.a		6	9.c	even	3	1
648.1.r.a		6	27.e	even	9	1
648.1.r.a		6	72.p	odd	6	1
648.1.r.a		6	216.r	odd	18	1
864.1.bd.a		6	36.h	even	6	1
864.1.bd.a		6	72.j	odd	6	1
864.1.bd.a		6	108.l	even	18	1
864.1.bd.a		6	216.x	odd	18	1
1944.1.r.a		6	9.d	odd	6	1
1944.1.r.a		6	27.f	odd	18	1
1944.1.r.a		6	72.l	even	6	1
1944.1.r.a		6	216.v	even	18	1
1944.1.r.b		6	3.b	odd	2	1
1944.1.r.b		6	24.f	even	2	1
1944.1.r.b		6	27.f	odd	18	1
1944.1.r.b		6	216.v	even	18	1
1944.1.r.c		6	1.a	even	1	1	trivial
1944.1.r.c		6	8.d	odd	2	1	CM
1944.1.r.c		6	27.e	even	9	1	inner
1944.1.r.c		6	216.r	odd	18	1	inner
1944.1.r.d		6	9.c	even	3	1
1944.1.r.d		6	27.e	even	9	1
1944.1.r.d		6	72.p	odd	6	1
1944.1.r.d		6	216.r	odd	18	1
2592.1.bd.a		6	36.f	odd	6	1
2592.1.bd.a		6	72.n	even	6	1
2592.1.bd.a		6	108.j	odd	18	1
2592.1.bd.a		6	216.t	even	18	1

This newform subspace can be constructed as the kernel of the linear operator $ T_{11}^{6} + 6T_{11}^{5} + 15T_{11}^{4} + 19T_{11}^{3} + 12T_{11}^{2} + 3T_{11} + 1 $ T11^6 + 6*T11^5 + 15*T11^4 + 19*T11^3 + 12*T11^2 + 3*T11 + 1 acting on $S_{1}^{\mathrm{new}}(1944, [\chi])$.

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

Level:	\( N \)	\(=\)	\( 1944 = 2^{3} \cdot 3^{5} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1944.r (of order \(18\), degree \(6\), not minimal)

\(f(q)\)	\(=\)	\( q - \zeta_{18}^{4} q^{2} + \zeta_{18}^{8} q^{4} + \zeta_{18}^{3} q^{8} +O(q^{10}) \) q - z^4 * q^2 + z^8 * q^4 + z^3 * q^8	\( q - \zeta_{18}^{4} q^{2} + \zeta_{18}^{8} q^{4} + \zeta_{18}^{3} q^{8} + (\zeta_{18}^{5} - 1) q^{11} - \zeta_{18}^{7} q^{16} + (\zeta_{18}^{7} + \zeta_{18}^{5}) q^{17} + (\zeta_{18}^{8} - \zeta_{18}^{7}) q^{19} + (\zeta_{18}^{4} + 1) q^{22} + \zeta_{18}^{4} q^{25} - \zeta_{18}^{2} q^{32} + (\zeta_{18}^{2} + 1) q^{34} + (\zeta_{18}^{3} - \zeta_{18}^{2}) q^{38} + (\zeta_{18} - 1) q^{41} + (\zeta_{18}^{8} + \zeta_{18}^{6}) q^{43} + ( - \zeta_{18}^{8} - \zeta_{18}^{4}) q^{44} + \zeta_{18}^{2} q^{49} - \zeta_{18}^{8} q^{50} + (\zeta_{18}^{7} - \zeta_{18}^{6}) q^{59} + \zeta_{18}^{6} q^{64} + ( - \zeta_{18}^{7} - \zeta_{18}^{3}) q^{67} + ( - \zeta_{18}^{6} - \zeta_{18}^{4}) q^{68} + (\zeta_{18}^{4} + \zeta_{18}^{2}) q^{73} + ( - \zeta_{18}^{7} + \zeta_{18}^{6}) q^{76} + ( - \zeta_{18}^{5} + \zeta_{18}^{4}) q^{82} + \zeta_{18}^{4} q^{83} + (\zeta_{18}^{3} + \zeta_{18}) q^{86} + (\zeta_{18}^{8} - \zeta_{18}^{3}) q^{88} - \zeta_{18}^{3} q^{89} + ( - \zeta_{18}^{5} + 1) q^{97} - \zeta_{18}^{6} q^{98} +O(q^{100}) \) q - z^4 * q^2 + z^8 * q^4 + z^3 * q^8 + (z^5 - 1) * q^11 - z^7 * q^16 + (z^7 + z^5) * q^17 + (z^8 - z^7) * q^19 + (z^4 + 1) * q^22 + z^4 * q^25 - z^2 * q^32 + (z^2 + 1) * q^34 + (z^3 - z^2) * q^38 + (z - 1) * q^41 + (z^8 + z^6) * q^43 + (-z^8 - z^4) * q^44 + z^2 * q^49 - z^8 * q^50 + (z^7 - z^6) * q^59 + z^6 * q^64 + (-z^7 - z^3) * q^67 + (-z^6 - z^4) * q^68 + (z^4 + z^2) * q^73 + (-z^7 + z^6) * q^76 + (-z^5 + z^4) * q^82 + z^4 * q^83 + (z^3 + z) * q^86 + (z^8 - z^3) * q^88 - z^3 * q^89 + (-z^5 + 1) * q^97 - z^6 * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 3 q^{8}+O(q^{10}) \) 6 * q + 3 * q^8	\( 6 q + 3 q^{8} - 6 q^{11} + 6 q^{22} + 6 q^{34} + 3 q^{38} - 6 q^{41} - 3 q^{43} + 3 q^{59} - 3 q^{64} - 3 q^{67} + 3 q^{68} - 3 q^{76} + 3 q^{86} - 3 q^{88} - 3 q^{89} + 6 q^{97} + 3 q^{98}+O(q^{100}) \) 6 * q + 3 * q^8 - 6 * q^11 + 6 * q^22 + 6 * q^34 + 3 * q^38 - 6 * q^41 - 3 * q^43 + 3 * q^59 - 3 * q^64 - 3 * q^67 + 3 * q^68 - 3 * q^76 + 3 * q^86 - 3 * q^88 - 3 * q^89 + 6 * q^97 + 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	1944.1.r.c

Level	$1944$
Weight	$1$
Character orbit	1944.r
Analytic conductor	$0.970$
Analytic rank	$0$
Dimension	$6$
Projective image	$D_{9}$
CM discriminant	-8
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(0.970182384559\)
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:	no (minimal twist has level 216)
Projective image:	\(D_{9}\)
Projective field:	Galois closure of 9.1.128536820158464.7

\(n\)	\(487\)	\(973\)	\(1217\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(\zeta_{18}^{8}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 379.1
Significant digits:
Format:

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.d	odd	2	1	CM by \(\Q(\sqrt{-2}) \)
27.e	even	9	1	inner
216.r	odd	18	1	inner

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