LMFDB - Newform orbit 1800.2.s.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1800,2,Mod(593,1800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1800, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 0, 2, 3]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1800.593");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1800.s (of order $4$, degree $2$, 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:	$14.3730723638$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 360)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $q$-expansion are expressed in terms of a basis $1,\beta_1,\beta_2,\beta_3$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q+O(q^{10}) $ q	$ q + 4 \beta_{2} q^{11} + (3 \beta_1 - 3) q^{13} - 4 \beta_1 q^{19} + ( - 2 \beta_{3} - 2 \beta_{2}) q^{23} + \beta_{3} q^{29} - 8 q^{31} + ( - 7 \beta_1 - 7) q^{37} - \beta_{2} q^{41} + (4 \beta_1 - 4) q^{43} + ( - 2 \beta_{3} + 2 \beta_{2}) q^{47} - 7 \beta_1 q^{49} + (6 \beta_{3} + 6 \beta_{2}) q^{53} - 8 \beta_{3} q^{59} - 12 q^{61} + (8 \beta_1 + 8) q^{67} + 4 \beta_{2} q^{71} + ( - 3 \beta_1 + 3) q^{73} + 8 \beta_1 q^{79} + (8 \beta_{3} + 8 \beta_{2}) q^{83} - 5 \beta_{3} q^{89} + ( - 5 \beta_1 - 5) q^{97}+O(q^{100}) $ q + 4b2 q^11 + (3b1 - 3) q^13 - 4b1 q^19 + (-2b3 - 2b2) * q^23 + b3 * q^29 - 8 * q^31 + (-7b1 - 7) q^37 - b2 * q^41 + (4b1 - 4) q^43 + (-2b3 + 2b2) * q^47 - 7b1 q^49 + (6b3 + 6b2) * q^53 - 8b3 q^59 - 12 * q^61 + (8b1 + 8) q^67 + 4b2 q^71 + (-3b1 + 3) q^73 + 8b1 q^79 + (8b3 + 8b2) * q^83 - 5b3 q^89 + (-5b1 - 5) q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q - 12 q^{13} - 32 q^{31} - 28 q^{37} - 16 q^{43} - 48 q^{61} + 32 q^{67} + 12 q^{73} - 20 q^{97}+O(q^{100}) $ 4 * q - 12 * q^13 - 32 * q^31 - 28 * q^37 - 16 * q^43 - 48 * q^61 + 32 * q^67 + 12 * q^73 - 20 * q^97

Display 100 coefficients 10 coefficients

Basis of coefficient ring

$\beta_{1}$	$=$	$ \zeta_{8}^{2} $ v^2
$\beta_{2}$	$=$	$ \zeta_{8}^{3} + \zeta_{8} $ v^3 + v
$\beta_{3}$	$=$	$ -\zeta_{8}^{3} + \zeta_{8} $ -v^3 + v

Display $\zeta_{8}^j$ in terms of $\beta_i$

$\zeta_{8}$	$=$	$ ( \beta_{3} + \beta_{2} ) / 2 $ (b3 + b2) / 2
$\zeta_{8}^{2}$	$=$	$ \beta_1 $ b1
$\zeta_{8}^{3}$	$=$	$ ( -\beta_{3} + \beta_{2} ) / 2 $ (-b3 + b2) / 2

Display $\beta_i$ in terms of $\zeta_{8}^j$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1800\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$1001$	$1351$
$\chi(n)$	$\beta_{1}$	$1$	$-1$	$1$

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} $

593.1

0.707107	−	0.707107i
−0.707107	+	0.707107i
−0.707107	−	0.707107i
0.707107	+	0.707107i

593.2

1457.1

1457.2

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
5.c	odd	4	1	inner
15.e	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1800.2.s.b		4
3.b	odd	2	1	inner	1800.2.s.b		4
4.b	odd	2	1		3600.2.w.d		4
5.b	even	2	1		360.2.s.a	✓	4
5.c	odd	4	1		360.2.s.a	✓	4
5.c	odd	4	1	inner	1800.2.s.b		4
12.b	even	2	1		3600.2.w.d		4
15.d	odd	2	1		360.2.s.a	✓	4
15.e	even	4	1		360.2.s.a	✓	4
15.e	even	4	1	inner	1800.2.s.b		4
20.d	odd	2	1		720.2.w.c		4
20.e	even	4	1		720.2.w.c		4
20.e	even	4	1		3600.2.w.d		4
40.e	odd	2	1		2880.2.w.f		4
40.f	even	2	1		2880.2.w.e		4
40.i	odd	4	1		2880.2.w.e		4
40.k	even	4	1		2880.2.w.f		4
60.h	even	2	1		720.2.w.c		4
60.l	odd	4	1		720.2.w.c		4
60.l	odd	4	1		3600.2.w.d		4
120.i	odd	2	1		2880.2.w.e		4
120.m	even	2	1		2880.2.w.f		4
120.q	odd	4	1		2880.2.w.f		4
120.w	even	4	1		2880.2.w.e		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
360.2.s.a	✓	4	5.b	even	2	1
360.2.s.a	✓	4	5.c	odd	4	1
360.2.s.a	✓	4	15.d	odd	2	1
360.2.s.a	✓	4	15.e	even	4	1
720.2.w.c		4	20.d	odd	2	1
720.2.w.c		4	20.e	even	4	1
720.2.w.c		4	60.h	even	2	1
720.2.w.c		4	60.l	odd	4	1
1800.2.s.b		4	1.a	even	1	1	trivial
1800.2.s.b		4	3.b	odd	2	1	inner
1800.2.s.b		4	5.c	odd	4	1	inner
1800.2.s.b		4	15.e	even	4	1	inner
2880.2.w.e		4	40.f	even	2	1
2880.2.w.e		4	40.i	odd	4	1
2880.2.w.e		4	120.i	odd	2	1
2880.2.w.e		4	120.w	even	4	1
2880.2.w.f		4	40.e	odd	2	1
2880.2.w.f		4	40.k	even	4	1
2880.2.w.f		4	120.m	even	2	1
2880.2.w.f		4	120.q	odd	4	1
3600.2.w.d		4	4.b	odd	2	1
3600.2.w.d		4	12.b	even	2	1
3600.2.w.d		4	20.e	even	4	1
3600.2.w.d		4	60.l	odd	4	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{7} $ T7 acting on $S_{2}^{\mathrm{new}}(1800, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{4} $ T^4
$3$	$ T^{4} $ T^4
$5$	$ T^{4} $ T^4
$7$	$ T^{4} $ T^4
$11$	$ (T^{2} + 32)^{2} $ (T^2 + 32)^2
$13$	$ (T^{2} + 6 T + 18)^{2} $ (T^2 + 6*T + 18)^2
$17$	$ T^{4} $ T^4
$19$	$ (T^{2} + 16)^{2} $ (T^2 + 16)^2
$23$	$ T^{4} + 256 $ T^4 + 256
$29$	$ (T^{2} - 2)^{2} $ (T^2 - 2)^2
$31$	$ (T + 8)^{4} $ (T + 8)^4
$37$	$ (T^{2} + 14 T + 98)^{2} $ (T^2 + 14*T + 98)^2
$41$	$ (T^{2} + 2)^{2} $ (T^2 + 2)^2
$43$	$ (T^{2} + 8 T + 32)^{2} $ (T^2 + 8*T + 32)^2
$47$	$ T^{4} + 256 $ T^4 + 256
$53$	$ T^{4} + 20736 $ T^4 + 20736
$59$	$ (T^{2} - 128)^{2} $ (T^2 - 128)^2
$61$	$ (T + 12)^{4} $ (T + 12)^4
$67$	$ (T^{2} - 16 T + 128)^{2} $ (T^2 - 16*T + 128)^2
$71$	$ (T^{2} + 32)^{2} $ (T^2 + 32)^2
$73$	$ (T^{2} - 6 T + 18)^{2} $ (T^2 - 6*T + 18)^2
$79$	$ (T^{2} + 64)^{2} $ (T^2 + 64)^2
$83$	$ T^{4} + 65536 $ T^4 + 65536
$89$	$ (T^{2} - 50)^{2} $ (T^2 - 50)^2
$97$	$ (T^{2} + 10 T + 50)^{2} $ (T^2 + 10*T + 50)^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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1800.s (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\( q+O(q^{10}) \) q	\( q + 4 \beta_{2} q^{11} + (3 \beta_1 - 3) q^{13} - 4 \beta_1 q^{19} + ( - 2 \beta_{3} - 2 \beta_{2}) q^{23} + \beta_{3} q^{29} - 8 q^{31} + ( - 7 \beta_1 - 7) q^{37} - \beta_{2} q^{41} + (4 \beta_1 - 4) q^{43} + ( - 2 \beta_{3} + 2 \beta_{2}) q^{47} - 7 \beta_1 q^{49} + (6 \beta_{3} + 6 \beta_{2}) q^{53} - 8 \beta_{3} q^{59} - 12 q^{61} + (8 \beta_1 + 8) q^{67} + 4 \beta_{2} q^{71} + ( - 3 \beta_1 + 3) q^{73} + 8 \beta_1 q^{79} + (8 \beta_{3} + 8 \beta_{2}) q^{83} - 5 \beta_{3} q^{89} + ( - 5 \beta_1 - 5) q^{97}+O(q^{100}) \) q + 4b2 q^11 + (3b1 - 3) q^13 - 4b1 q^19 + (-2b3 - 2b2) * q^23 + b3 * q^29 - 8 * q^31 + (-7b1 - 7) q^37 - b2 * q^41 + (4b1 - 4) q^43 + (-2b3 + 2b2) * q^47 - 7b1 q^49 + (6b3 + 6b2) * q^53 - 8b3 q^59 - 12 * q^61 + (8b1 + 8) q^67 + 4b2 q^71 + (-3b1 + 3) q^73 + 8b1 q^79 + (8b3 + 8b2) * q^83 - 5b3 q^89 + (-5b1 - 5) q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q - 12 q^{13} - 32 q^{31} - 28 q^{37} - 16 q^{43} - 48 q^{61} + 32 q^{67} + 12 q^{73} - 20 q^{97}+O(q^{100}) \) 4 * q - 12 * q^13 - 32 * q^31 - 28 * q^37 - 16 * q^43 - 48 * q^61 + 32 * q^67 + 12 * q^73 - 20 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more