LMFDB - Newform orbit 1800.4.a.bf

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1800,4,Mod(1,1800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1800, 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("1800.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1800.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:	$106.203438010$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 600)
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 + 19 q^{7}+O(q^{10}) $

$ q + 19 q^{7} - 22 q^{11} - q^{13} - 58 q^{17} - 53 q^{19} + 58 q^{23} - 22 q^{29} - 35 q^{31} + 270 q^{37} + 468 q^{41} + 431 q^{43} - 230 q^{47} + 18 q^{49} - 446 q^{59} + 127 q^{61} + 811 q^{67} - 36 q^{71} - 522 q^{73} - 418 q^{77} + 1368 q^{79} - 1138 q^{83} - 144 q^{89} - 19 q^{91} + 1079 q^{97}+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

19.0000

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	1800.4.a.bf		1
3.b	odd	2	1		600.4.a.g	✓	1
5.b	even	2	1		1800.4.a.g		1
5.c	odd	4	2		1800.4.f.g		2
12.b	even	2	1		1200.4.a.x		1
15.d	odd	2	1		600.4.a.j	yes	1
15.e	even	4	2		600.4.f.h		2
60.h	even	2	1		1200.4.a.n		1
60.l	odd	4	2		1200.4.f.f		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
600.4.a.g	✓	1	3.b	odd	2	1
600.4.a.j	yes	1	15.d	odd	2	1
600.4.f.h		2	15.e	even	4	2
1200.4.a.n		1	60.h	even	2	1
1200.4.a.x		1	12.b	even	2	1
1200.4.f.f		2	60.l	odd	4	2
1800.4.a.g		1	5.b	even	2	1
1800.4.a.bf		1	1.a	even	1	1	trivial
1800.4.f.g		2	5.c	odd	4	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(\Gamma_0(1800))$:

$ T_{7} - 19 $

$ T_{11} + 22 $

$ T_{17} + 58 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ T $ T
$5$	$ T $ T
$7$	$ T - 19 $ T - 19
$11$	$ T + 22 $ T + 22
$13$	$ T + 1 $ T + 1
$17$	$ T + 58 $ T + 58
$19$	$ T + 53 $ T + 53
$23$	$ T - 58 $ T - 58
$29$	$ T + 22 $ T + 22
$31$	$ T + 35 $ T + 35
$37$	$ T - 270 $ T - 270
$41$	$ T - 468 $ T - 468
$43$	$ T - 431 $ T - 431
$47$	$ T + 230 $ T + 230
$53$	$ T $ T
$59$	$ T + 446 $ T + 446
$61$	$ T - 127 $ T - 127
$67$	$ T - 811 $ T - 811
$71$	$ T + 36 $ T + 36
$73$	$ T + 522 $ T + 522
$79$	$ T - 1368 $ T - 1368
$83$	$ T + 1138 $ T + 1138
$89$	$ T + 144 $ T + 144
$97$	$ T - 1079 $ T - 1079
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Overview	Random
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Classical	Maass
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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:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1800.a (trivial)

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