LMFDB - Newform orbit 1200.4.a.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1200,4,Mod(1,1200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1200, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1200.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1200 = 2^{4} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1200.a (trivial)

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:	yes
Analytic conductor:	$70.8022920069$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 6)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$ q - 3 q^{3} - 16 q^{7} + 9 q^{9}+O(q^{10}) $

$ q - 3 q^{3} - 16 q^{7} + 9 q^{9} - 12 q^{11} - 38 q^{13} + 126 q^{17} - 20 q^{19} + 48 q^{21} + 168 q^{23} - 27 q^{27} + 30 q^{29} + 88 q^{31} + 36 q^{33} - 254 q^{37} + 114 q^{39} + 42 q^{41} - 52 q^{43} - 96 q^{47} - 87 q^{49} - 378 q^{51} - 198 q^{53} + 60 q^{57} + 660 q^{59} - 538 q^{61} - 144 q^{63} + 884 q^{67} - 504 q^{69} - 792 q^{71} - 218 q^{73} + 192 q^{77} + 520 q^{79} + 81 q^{81} - 492 q^{83} - 90 q^{87} + 810 q^{89} + 608 q^{91} - 264 q^{93} - 1154 q^{97} - 108 q^{99}+O(q^{100}) $

Display 100 coefficients 10 coefficients

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} $

1.1

−3.00000

−16.0000

9.00000

Atkin-Lehner signs

$ p $	Sign
$2$	$-1$
$3$	$1$
$5$	$1$

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1200.4.a.b		1
4.b	odd	2	1		150.4.a.i		1
5.b	even	2	1		48.4.a.c		1
5.c	odd	4	2		1200.4.f.j		2
12.b	even	2	1		450.4.a.h		1
15.d	odd	2	1		144.4.a.c		1
20.d	odd	2	1		6.4.a.a	✓	1
20.e	even	4	2		150.4.c.d		2
35.c	odd	2	1		2352.4.a.e		1
40.e	odd	2	1		192.4.a.i		1
40.f	even	2	1		192.4.a.c		1
60.h	even	2	1		18.4.a.a		1
60.l	odd	4	2		450.4.c.e		2
80.k	odd	4	2		768.4.d.n		2
80.q	even	4	2		768.4.d.c		2
120.i	odd	2	1		576.4.a.r		1
120.m	even	2	1		576.4.a.q		1
140.c	even	2	1		294.4.a.e		1
140.p	odd	6	2		294.4.e.h		2
140.s	even	6	2		294.4.e.g		2
180.n	even	6	2		162.4.c.c		2
180.p	odd	6	2		162.4.c.f		2
220.g	even	2	1		726.4.a.f		1
260.g	odd	2	1		1014.4.a.g		1
260.u	even	4	2		1014.4.b.d		2
340.d	odd	2	1		1734.4.a.d		1
380.d	even	2	1		2166.4.a.i		1
420.o	odd	2	1		882.4.a.n		1
420.ba	even	6	2		882.4.g.i		2
420.be	odd	6	2		882.4.g.f		2
660.g	odd	2	1		2178.4.a.e		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
6.4.a.a	✓	1	20.d	odd	2	1
18.4.a.a		1	60.h	even	2	1
48.4.a.c		1	5.b	even	2	1
144.4.a.c		1	15.d	odd	2	1
150.4.a.i		1	4.b	odd	2	1
150.4.c.d		2	20.e	even	4	2
162.4.c.c		2	180.n	even	6	2
162.4.c.f		2	180.p	odd	6	2
192.4.a.c		1	40.f	even	2	1
192.4.a.i		1	40.e	odd	2	1
294.4.a.e		1	140.c	even	2	1
294.4.e.g		2	140.s	even	6	2
294.4.e.h		2	140.p	odd	6	2
450.4.a.h		1	12.b	even	2	1
450.4.c.e		2	60.l	odd	4	2
576.4.a.q		1	120.m	even	2	1
576.4.a.r		1	120.i	odd	2	1
726.4.a.f		1	220.g	even	2	1
768.4.d.c		2	80.q	even	4	2
768.4.d.n		2	80.k	odd	4	2
882.4.a.n		1	420.o	odd	2	1
882.4.g.f		2	420.be	odd	6	2
882.4.g.i		2	420.ba	even	6	2
1014.4.a.g		1	260.g	odd	2	1
1014.4.b.d		2	260.u	even	4	2
1200.4.a.b		1	1.a	even	1	1	trivial
1200.4.f.j		2	5.c	odd	4	2
1734.4.a.d		1	340.d	odd	2	1
2166.4.a.i		1	380.d	even	2	1
2178.4.a.e		1	660.g	odd	2	1
2352.4.a.e		1	35.c	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(\Gamma_0(1200))$:

$ T_{7} + 16 $

$ T_{11} + 12 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ T + 3 $ T + 3
$5$	$ T $ T
$7$	$ T + 16 $ T + 16
$11$	$ T + 12 $ T + 12
$13$	$ T + 38 $ T + 38
$17$	$ T - 126 $ T - 126
$19$	$ T + 20 $ T + 20
$23$	$ T - 168 $ T - 168
$29$	$ T - 30 $ T - 30
$31$	$ T - 88 $ T - 88
$37$	$ T + 254 $ T + 254
$41$	$ T - 42 $ T - 42
$43$	$ T + 52 $ T + 52
$47$	$ T + 96 $ T + 96
$53$	$ T + 198 $ T + 198
$59$	$ T - 660 $ T - 660
$61$	$ T + 538 $ T + 538
$67$	$ T - 884 $ T - 884
$71$	$ T + 792 $ T + 792
$73$	$ T + 218 $ T + 218
$79$	$ T - 520 $ T - 520
$83$	$ T + 492 $ T + 492
$89$	$ T - 810 $ T - 810
$97$	$ T + 1154 $ T + 1154
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:	\( N \)	\(=\)	\( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1200.a (trivial)

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more