LMFDB - Newform orbit 3648.1.cr.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3648,1,Mod(833,3648)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3648, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([0, 0, 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("3648.833");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3648 = 2^{6} \cdot 3 \cdot 19 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3648.cr (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:	$1.82058916609$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 228)
Projective image:	$D_{9}$
Projective field:	Galois closure of 9.1.88042790804544.1

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3648\mathbb{Z}\right)^\times$.

$n$	$1217$	$1921$	$2053$	$2623$
$\chi(n)$	$-1$	$\zeta_{18}^{2}$	$1$	$1$

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} $

833.1

−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

−0.939693

−

1.62760i

0.766044

−

0.642788i

1601.1

0.173648

−

0.984808i

0.173648

0.300767i

−0.939693

−

0.342020i

1985.1

0.766044

−

0.642788i

0.766044

−

1.32683i

0.173648

−

0.984808i

2753.1

0.766044

0.642788i

0.766044

1.32683i

0.173648

0.984808i

3329.1

0.173648

0.984808i

0.173648

−

0.300767i

−0.939693

0.342020i

3521.1

−0.939693

−

0.342020i

−0.939693

1.62760i

0.766044

0.642788i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by $\Q(\sqrt{-3}) $
19.e	even	9	1	inner
57.l	odd	18	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3648.1.cr.a		6
3.b	odd	2	1	CM	3648.1.cr.a		6
4.b	odd	2	1		3648.1.cr.b		6
8.b	even	2	1		912.1.cb.a		6
8.d	odd	2	1		228.1.s.a	✓	6
12.b	even	2	1		3648.1.cr.b		6
19.e	even	9	1	inner	3648.1.cr.a		6
24.f	even	2	1		228.1.s.a	✓	6
24.h	odd	2	1		912.1.cb.a		6
57.l	odd	18	1	inner	3648.1.cr.a		6
76.l	odd	18	1		3648.1.cr.b		6
152.t	even	18	1		912.1.cb.a		6
152.u	odd	18	1		228.1.s.a	✓	6
228.v	even	18	1		3648.1.cr.b		6
456.bh	odd	18	1		912.1.cb.a		6
456.bu	even	18	1		228.1.s.a	✓	6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
228.1.s.a	✓	6	8.d	odd	2	1
228.1.s.a	✓	6	24.f	even	2	1
228.1.s.a	✓	6	152.u	odd	18	1
228.1.s.a	✓	6	456.bu	even	18	1
912.1.cb.a		6	8.b	even	2	1
912.1.cb.a		6	24.h	odd	2	1
912.1.cb.a		6	152.t	even	18	1
912.1.cb.a		6	456.bh	odd	18	1
3648.1.cr.a		6	1.a	even	1	1	trivial
3648.1.cr.a		6	3.b	odd	2	1	CM
3648.1.cr.a		6	19.e	even	9	1	inner
3648.1.cr.a		6	57.l	odd	18	1	inner
3648.1.cr.b		6	4.b	odd	2	1
3648.1.cr.b		6	12.b	even	2	1
3648.1.cr.b		6	76.l	odd	18	1
3648.1.cr.b		6	228.v	even	18	1

This newform subspace can be constructed as the kernel of the linear operator $ T_{7}^{6} + 3T_{7}^{4} - 2T_{7}^{3} + 9T_{7}^{2} - 3T_{7} + 1 $ T7^6 + 3*T7^4 - 2*T7^3 + 9*T7^2 - 3*T7 + 1 acting on $S_{1}^{\mathrm{new}}(3648, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{6} $ T^6
$3$	$ T^{6} + T^{3} + 1 $ T^6 + T^3 + 1
$5$	$ T^{6} $ T^6
$7$	$ T^{6} + 3 T^{4} - 2 T^{3} + 9 T^{2} + \cdots + 1 $ T^6 + 3T^4 - 2T^3 + 9T^2 - 3T + 1
$11$	$ T^{6} $ T^6
$13$	$ T^{6} - 3 T^{5} + 6 T^{4} - 8 T^{3} + \cdots + 1 $ T^6 - 3T^5 + 6T^4 - 8T^3 + 12T^2 - 6*T + 1
$17$	$ T^{6} $ T^6
$19$	$ (T^{2} + T + 1)^{3} $ (T^2 + T + 1)^3
$23$	$ T^{6} $ T^6
$29$	$ T^{6} $ T^6
$31$	$ T^{6} + 3 T^{4} - 2 T^{3} + 9 T^{2} + \cdots + 1 $ T^6 + 3T^4 - 2T^3 + 9T^2 - 3T + 1
$37$	$ (T^{3} - 3 T - 1)^{2} $ (T^3 - 3*T - 1)^2
$41$	$ T^{6} $ T^6
$43$	$ T^{6} + 3 T^{5} + 6 T^{4} + 8 T^{3} + \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} $ T^6
$61$	$ T^{6} + 6 T^{5} + 15 T^{4} + 19 T^{3} + \cdots + 1 $ T^6 + 6T^5 + 15T^4 + 19T^3 + 12T^2 + 3*T + 1
$67$	$ T^{6} + 3 T^{5} + 6 T^{4} + 8 T^{3} + \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^{5} + 6 T^{4} + 8 T^{3} + \cdots + 1 $ T^6 + 3T^5 + 6T^4 + 8T^3 + 12T^2 + 6*T + 1
$79$	$ T^{6} + 6 T^{5} + 15 T^{4} + 19 T^{3} + \cdots + 1 $ T^6 + 6T^5 + 15T^4 + 19T^3 + 12T^2 + 3*T + 1
$83$	$ T^{6} $ T^6
$89$	$ T^{6} $ T^6
$97$	$ 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 $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 3648 = 2^{6} \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3648.cr (of order \(18\), degree \(6\), not minimal)

\(f(q)\)	\(=\)	\( q + \zeta_{18}^{4} q^{3} + ( - \zeta_{18}^{2} + \zeta_{18}) q^{7} + \zeta_{18}^{8} q^{9}+O(q^{10}) \) q + z^4 * q^3 + (-z^2 + z) * q^7 + z^8 * q^9	\( q + \zeta_{18}^{4} q^{3} + ( - \zeta_{18}^{2} + \zeta_{18}) q^{7} + \zeta_{18}^{8} q^{9} + (\zeta_{18}^{7} + \zeta_{18}^{3}) q^{13} + \zeta_{18}^{6} q^{19} + ( - \zeta_{18}^{6} + \zeta_{18}^{5}) q^{21} - \zeta_{18}^{5} q^{25} - \zeta_{18}^{3} q^{27} + (\zeta_{18}^{7} + \zeta_{18}^{5}) q^{31} + ( - \zeta_{18}^{8} + \zeta_{18}) q^{37} + (\zeta_{18}^{7} - \zeta_{18}^{2}) q^{39} + (\zeta_{18}^{8} + \zeta_{18}^{6}) q^{43} + (\zeta_{18}^{4} - \zeta_{18}^{3} + \zeta_{18}^{2}) q^{49} - \zeta_{18} q^{57} + ( - \zeta_{18}^{4} - 1) q^{61} + (\zeta_{18} - 1) q^{63} + (\zeta_{18}^{4} - \zeta_{18}^{3}) q^{67} + (\zeta_{18}^{6} + \zeta_{18}^{2}) q^{73} + q^{75} + ( - \zeta_{18}^{8} - 1) q^{79} - \zeta_{18}^{7} q^{81} + (\zeta_{18}^{8} - \zeta_{18}^{5} + \zeta_{18}^{4} + 1) q^{91} + ( - \zeta_{18}^{2} - 1) q^{93} + \zeta_{18} q^{97} +O(q^{100}) \) q + z^4 * q^3 + (-z^2 + z) * q^7 + z^8 * q^9 + (z^7 + z^3) * q^13 + z^6 * q^19 + (-z^6 + z^5) * q^21 - z^5 * q^25 - z^3 * q^27 + (z^7 + z^5) * q^31 + (-z^8 + z) * q^37 + (z^7 - z^2) * q^39 + (z^8 + z^6) * q^43 + (z^4 - z^3 + z^2) * q^49 - z * q^57 + (-z^4 - 1) * q^61 + (z - 1) * q^63 + (z^4 - z^3) * q^67 + (z^6 + z^2) * q^73 + q^75 + (-z^8 - 1) * q^79 - z^7 * q^81 + (z^8 - z^5 + z^4 + 1) * q^91 + (-z^2 - 1) * q^93 + z * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q+O(q^{10}) \) 6 * q	\( 6 q + 3 q^{13} - 3 q^{19} + 3 q^{21} - 3 q^{27} - 3 q^{43} - 3 q^{49} - 6 q^{61} - 6 q^{63} - 3 q^{67} - 3 q^{73} + 6 q^{75} - 6 q^{79} + 6 q^{91} - 6 q^{93}+O(q^{100}) \) 6 * q + 3 * q^13 - 3 * q^19 + 3 * q^21 - 3 * q^27 - 3 * q^43 - 3 * q^49 - 6 * q^61 - 6 * q^63 - 3 * q^67 - 3 * q^73 + 6 * q^75 - 6 * q^79 + 6 * q^91 - 6 * q^93

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	3648.1.cr.a

Level	$3648$
Weight	$1$
Character orbit	3648.cr
Analytic conductor	$1.821$
Analytic rank	$0$
Dimension	$6$
Projective image	$D_{9}$
CM discriminant	-3
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(1.82058916609\)
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:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 228)
Projective image:	\(D_{9}\)
Projective field:	Galois closure of 9.1.88042790804544.1

\(n\)	\(1217\)	\(1921\)	\(2053\)	\(2623\)
\(\chi(n)\)	\(-1\)	\(\zeta_{18}^{2}\)	\(1\)	\(1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 833.1
Significant digits:
Format:

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by \(\Q(\sqrt{-3}) \)
19.e	even	9	1	inner
57.l	odd	18	1	inner

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