LMFDB - Newform orbit 2450.4.a.ba

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2450,4,Mod(1,2450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2450, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2450.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$2450 = 2 \cdot 5^{2} \cdot 7^{2}$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	2450.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:	$144.554679514$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 70)
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 + 2 q^{2} - 3 q^{3} + 4 q^{4} - 6 q^{6} + 8 q^{8} - 18 q^{9} - 17 q^{11} - 12 q^{12} - 81 q^{13} + 16 q^{16} - 91 q^{17} - 36 q^{18} - 102 q^{19} - 34 q^{22} + 90 q^{23} - 24 q^{24} - 162 q^{26} + 135 q^{27}+ \cdots + 306 q^{99}+O(q^{100})

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

2.00000

−3.00000

4.00000

−6.00000

8.00000

−18.0000

Atkin-Lehner signs

$p$	Sign
$2$	$-1$
$5$	$+1$
$7$	$-1$

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	2450.4.a.ba		1
5.b	even	2	1		490.4.a.f		1
7.b	odd	2	1		350.4.a.t		1
35.c	odd	2	1		70.4.a.b	✓	1
35.f	even	4	2		350.4.c.j		2
35.i	odd	6	2		490.4.e.p		2
35.j	even	6	2		490.4.e.l		2
105.g	even	2	1		630.4.a.m		1
140.c	even	2	1		560.4.a.k		1
280.c	odd	2	1		2240.4.a.w		1
280.n	even	2	1		2240.4.a.p		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
70.4.a.b	✓	1	35.c	odd	2	1
350.4.a.t		1	7.b	odd	2	1
350.4.c.j		2	35.f	even	4	2
490.4.a.f		1	5.b	even	2	1
490.4.e.l		2	35.j	even	6	2
490.4.e.p		2	35.i	odd	6	2
560.4.a.k		1	140.c	even	2	1
630.4.a.m		1	105.g	even	2	1
2240.4.a.p		1	280.n	even	2	1
2240.4.a.w		1	280.c	odd	2	1
2450.4.a.ba		1	1.a	even	1	1	trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(\Gamma_0(2450))$ :

T_{3} + 3

T_{11} + 17

T_{19} + 102

T_{23} - 90

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T - 2$ T - 2
$3$	$T + 3$ T + 3
$5$	$T$ T
$7$	$T$ T
$11$	$T + 17$ T + 17
$13$	$T + 81$ T + 81
$17$	$T + 91$ T + 91
$19$	$T + 102$ T + 102
$23$	$T - 90$ T - 90
$29$	$T + 129$ T + 129
$31$	$T + 116$ T + 116
$37$	$T + 314$ T + 314
$41$	$T - 124$ T - 124
$43$	$T - 434$ T - 434
$47$	$T - 497$ T - 497
$53$	$T - 584$ T - 584
$59$	$T - 332$ T - 332
$61$	$T + 220$ T + 220
$67$	$T + 384$ T + 384
$71$	$T + 664$ T + 664
$73$	$T - 230$ T - 230
$79$	$T - 361$ T - 361
$83$	$T - 1172$ T - 1172
$89$	$T + 40$ T + 40
$97$	$T + 175$ T + 175
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