LMFDB - Newform orbit 560.4.a.k

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [560,4,Mod(1,560)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(560, 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("560.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$560 = 2^{4} \cdot 5 \cdot 7$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	560.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:	$33.0410696032$
Analytic rank:	$1$
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 + 3 q^{3} + 5 q^{5} + 7 q^{7} - 18 q^{9} + 17 q^{11} - 81 q^{13} + 15 q^{15} - 91 q^{17} - 102 q^{19} + 21 q^{21} + 90 q^{23} + 25 q^{25} - 135 q^{27} - 129 q^{29} - 116 q^{31} + 51 q^{33} + 35 q^{35}+ \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

3.00000

5.00000

7.00000

−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	560.4.a.k		1
4.b	odd	2	1		70.4.a.b	✓	1
8.b	even	2	1		2240.4.a.p		1
8.d	odd	2	1		2240.4.a.w		1
12.b	even	2	1		630.4.a.m		1
20.d	odd	2	1		350.4.a.t		1
20.e	even	4	2		350.4.c.j		2
28.d	even	2	1		490.4.a.f		1
28.f	even	6	2		490.4.e.l		2
28.g	odd	6	2		490.4.e.p		2
140.c	even	2	1		2450.4.a.ba		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
70.4.a.b	✓	1	4.b	odd	2	1
350.4.a.t		1	20.d	odd	2	1
350.4.c.j		2	20.e	even	4	2
490.4.a.f		1	28.d	even	2	1
490.4.e.l		2	28.f	even	6	2
490.4.e.p		2	28.g	odd	6	2
560.4.a.k		1	1.a	even	1	1	trivial
630.4.a.m		1	12.b	even	2	1
2240.4.a.p		1	8.b	even	2	1
2240.4.a.w		1	8.d	odd	2	1
2450.4.a.ba		1	140.c	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(\Gamma_0(560))$ :

T_{3} - 3

T_{11} - 17

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T$ T
$3$	$T - 3$ T - 3
$5$	$T - 5$ T - 5
$7$	$T - 7$ T - 7
$11$	$T - 17$ 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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