LMFDB - Newform orbit 72.6.a.d

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [72,6,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, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("72.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$N$	$=$	$72 = 2^{3} \cdot 3^{2}$
Weight:	$k$	$=$	$6$
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:	$11.5476350265$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	yes
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 + 16 q^{5} + 12 q^{7} + 448 q^{11} - 206 q^{13} + 1952 q^{17} + 1064 q^{19} + 3712 q^{23} - 2869 q^{25} + 4080 q^{29} + 5324 q^{31} + 192 q^{35} - 9690 q^{37} - 9120 q^{41} + 16552 q^{43} - 14208 q^{47}+ \cdots + 76046 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

16.0000

12.0000

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	72.6.a.d	yes	1
3.b	odd	2	1		72.6.a.c	✓	1
4.b	odd	2	1		144.6.a.h		1
8.b	even	2	1		576.6.a.n		1
8.d	odd	2	1		576.6.a.m		1
12.b	even	2	1		144.6.a.e		1
24.f	even	2	1		576.6.a.w		1
24.h	odd	2	1		576.6.a.x		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
72.6.a.c	✓	1	3.b	odd	2	1
72.6.a.d	yes	1	1.a	even	1	1	trivial
144.6.a.e		1	12.b	even	2	1
144.6.a.h		1	4.b	odd	2	1
576.6.a.m		1	8.d	odd	2	1
576.6.a.n		1	8.b	even	2	1
576.6.a.w		1	24.f	even	2	1
576.6.a.x		1	24.h	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{5} - 16$ T5 - 16 acting on $S_{6}^{\mathrm{new}}(\Gamma_0(72))$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T$ T
$3$	$T$ T
$5$	$T - 16$ T - 16
$7$	$T - 12$ T - 12
$11$	$T - 448$ T - 448
$13$	$T + 206$ T + 206
$17$	$T - 1952$ T - 1952
$19$	$T - 1064$ T - 1064
$23$	$T - 3712$ T - 3712
$29$	$T - 4080$ T - 4080
$31$	$T - 5324$ T - 5324
$37$	$T + 9690$ T + 9690
$41$	$T + 9120$ T + 9120
$43$	$T - 16552$ T - 16552
$47$	$T + 14208$ T + 14208
$53$	$T + 21776$ T + 21776
$59$	$T + 31616$ T + 31616
$61$	$T + 13154$ T + 13154
$67$	$T - 27056$ T - 27056
$71$	$T - 9728$ T - 9728
$73$	$T - 9046$ T - 9046
$79$	$T + 58292$ T + 58292
$83$	$T - 86336$ T - 86336
$89$	$T - 75072$ T - 75072
$97$	$T - 76046$ T - 76046
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