LMFDB - Newform orbit 1800.4.a.o

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 120)
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 - 4 q^{7}+O(q^{10}) $

$ q - 4 q^{7} + 28 q^{11} + 16 q^{13} + 108 q^{17} + 32 q^{19} - 28 q^{23} + 238 q^{29} - 180 q^{31} + 40 q^{37} - 422 q^{41} - 276 q^{43} + 60 q^{47} - 327 q^{49} + 220 q^{53} + 804 q^{59} - 358 q^{61} + 884 q^{67} + 64 q^{71} + 152 q^{73} - 112 q^{77} - 932 q^{79} - 1292 q^{83} + 1146 q^{89} - 64 q^{91} - 824 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

−4.00000

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	1800.4.a.o		1
3.b	odd	2	1		600.4.a.k		1
5.b	even	2	1		1800.4.a.u		1
5.c	odd	4	2		360.4.f.a		2
12.b	even	2	1		1200.4.a.l		1
15.d	odd	2	1		600.4.a.f		1
15.e	even	4	2		120.4.f.c	✓	2
20.e	even	4	2		720.4.f.b		2
60.h	even	2	1		1200.4.a.z		1
60.l	odd	4	2		240.4.f.e		2
120.q	odd	4	2		960.4.f.a		2
120.w	even	4	2		960.4.f.b		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
120.4.f.c	✓	2	15.e	even	4	2
240.4.f.e		2	60.l	odd	4	2
360.4.f.a		2	5.c	odd	4	2
600.4.a.f		1	15.d	odd	2	1
600.4.a.k		1	3.b	odd	2	1
720.4.f.b		2	20.e	even	4	2
960.4.f.a		2	120.q	odd	4	2
960.4.f.b		2	120.w	even	4	2
1200.4.a.l		1	12.b	even	2	1
1200.4.a.z		1	60.h	even	2	1
1800.4.a.o		1	1.a	even	1	1	trivial
1800.4.a.u		1	5.b	even	2	1

Hecke kernels

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

$ T_{7} + 4 $

$ T_{11} - 28 $

$ T_{17} - 108 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ T $ T
$5$	$ T $ T
$7$	$ T + 4 $ T + 4
$11$	$ T - 28 $ T - 28
$13$	$ T - 16 $ T - 16
$17$	$ T - 108 $ T - 108
$19$	$ T - 32 $ T - 32
$23$	$ T + 28 $ T + 28
$29$	$ T - 238 $ T - 238
$31$	$ T + 180 $ T + 180
$37$	$ T - 40 $ T - 40
$41$	$ T + 422 $ T + 422
$43$	$ T + 276 $ T + 276
$47$	$ T - 60 $ T - 60
$53$	$ T - 220 $ T - 220
$59$	$ T - 804 $ T - 804
$61$	$ T + 358 $ T + 358
$67$	$ T - 884 $ T - 884
$71$	$ T - 64 $ T - 64
$73$	$ T - 152 $ T - 152
$79$	$ T + 932 $ T + 932
$83$	$ T + 1292 $ T + 1292
$89$	$ T - 1146 $ T - 1146
$97$	$ T + 824 $ T + 824
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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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