LMFDB - Newform orbit 72.4.a.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [72,4,Mod(1,72)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(72, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("72.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$72 = 2^{3} \cdot 3^{2}$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	72.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:	$4.24813752041$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 24)
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 - 14 q^{5} - 24 q^{7} + 28 q^{11} - 74 q^{13} - 82 q^{17} + 92 q^{19} - 8 q^{23} + 71 q^{25} + 138 q^{29} + 80 q^{31} + 336 q^{35} + 30 q^{37} - 282 q^{41} + 4 q^{43} - 240 q^{47} + 233 q^{49} + 130 q^{53}+ \cdots + 866 q^{97}+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

−14.0000

−24.0000

Atkin-Lehner signs

$p$	Sign
$2$	$+1$
$3$	$-1$

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	72.4.a.b		1
3.b	odd	2	1		24.4.a.a	✓	1
4.b	odd	2	1		144.4.a.b		1
5.b	even	2	1		1800.4.a.bg		1
5.c	odd	4	2		1800.4.f.q		2
8.b	even	2	1		576.4.a.u		1
8.d	odd	2	1		576.4.a.v		1
9.c	even	3	2		648.4.i.k		2
9.d	odd	6	2		648.4.i.b		2
12.b	even	2	1		48.4.a.b		1
15.d	odd	2	1		600.4.a.h		1
15.e	even	4	2		600.4.f.b		2
21.c	even	2	1		1176.4.a.a		1
24.f	even	2	1		192.4.a.g		1
24.h	odd	2	1		192.4.a.a		1
48.i	odd	4	2		768.4.d.o		2
48.k	even	4	2		768.4.d.b		2
60.h	even	2	1		1200.4.a.u		1
60.l	odd	4	2		1200.4.f.p		2
84.h	odd	2	1		2352.4.a.w		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
24.4.a.a	✓	1	3.b	odd	2	1
48.4.a.b		1	12.b	even	2	1
72.4.a.b		1	1.a	even	1	1	trivial
144.4.a.b		1	4.b	odd	2	1
192.4.a.a		1	24.h	odd	2	1
192.4.a.g		1	24.f	even	2	1
576.4.a.u		1	8.b	even	2	1
576.4.a.v		1	8.d	odd	2	1
600.4.a.h		1	15.d	odd	2	1
600.4.f.b		2	15.e	even	4	2
648.4.i.b		2	9.d	odd	6	2
648.4.i.k		2	9.c	even	3	2
768.4.d.b		2	48.k	even	4	2
768.4.d.o		2	48.i	odd	4	2
1176.4.a.a		1	21.c	even	2	1
1200.4.a.u		1	60.h	even	2	1
1200.4.f.p		2	60.l	odd	4	2
1800.4.a.bg		1	5.b	even	2	1
1800.4.f.q		2	5.c	odd	4	2
2352.4.a.w		1	84.h	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{5} + 14$ T5 + 14 acting on $S_{4}^{\mathrm{new}}(\Gamma_0(72))$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T$ T
$3$	$T$ T
$5$	$T + 14$ T + 14
$7$	$T + 24$ T + 24
$11$	$T - 28$ T - 28
$13$	$T + 74$ T + 74
$17$	$T + 82$ T + 82
$19$	$T - 92$ T - 92
$23$	$T + 8$ T + 8
$29$	$T - 138$ T - 138
$31$	$T - 80$ T - 80
$37$	$T - 30$ T - 30
$41$	$T + 282$ T + 282
$43$	$T - 4$ T - 4
$47$	$T + 240$ T + 240
$53$	$T - 130$ T - 130
$59$	$T + 596$ T + 596
$61$	$T + 218$ T + 218
$67$	$T + 436$ T + 436
$71$	$T + 856$ T + 856
$73$	$T + 998$ T + 998
$79$	$T + 32$ T + 32
$83$	$T - 1508$ T - 1508
$89$	$T - 246$ T - 246
$97$	$T - 866$ T - 866
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