LMFDB - Newform orbit 220.2.a.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [220,2,Mod(1,220)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(220, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("220.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$220 = 2^{2} \cdot 5 \cdot 11$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	220.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:	$1.75670884447$
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 + 2 q^{3} + q^{5} + q^{9} + q^{11} + 2 q^{15} - 4 q^{17} - 4 q^{19} + 6 q^{23} + q^{25} - 4 q^{27} + 2 q^{29} + 2 q^{33} - 6 q^{37} - 10 q^{41} + 4 q^{43} + q^{45} + 10 q^{47} - 7 q^{49} - 8 q^{51}+ \cdots + 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

1.00000

Atkin-Lehner signs

$p$	Sign
$2$	$-1$
$5$	$-1$
$11$	$-1$

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	220.2.a.b	✓	1
3.b	odd	2	1		1980.2.a.b		1
4.b	odd	2	1		880.2.a.b		1
5.b	even	2	1		1100.2.a.a		1
5.c	odd	4	2		1100.2.b.b		2
8.b	even	2	1		3520.2.a.c		1
8.d	odd	2	1		3520.2.a.bf		1
11.b	odd	2	1		2420.2.a.g		1
12.b	even	2	1		7920.2.a.l		1
15.d	odd	2	1		9900.2.a.n		1
15.e	even	4	2		9900.2.c.e		2
20.d	odd	2	1		4400.2.a.z		1
20.e	even	4	2		4400.2.b.c		2
44.c	even	2	1		9680.2.a.d		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
220.2.a.b	✓	1	1.a	even	1	1	trivial
880.2.a.b		1	4.b	odd	2	1
1100.2.a.a		1	5.b	even	2	1
1100.2.b.b		2	5.c	odd	4	2
1980.2.a.b		1	3.b	odd	2	1
2420.2.a.g		1	11.b	odd	2	1
3520.2.a.c		1	8.b	even	2	1
3520.2.a.bf		1	8.d	odd	2	1
4400.2.a.z		1	20.d	odd	2	1
4400.2.b.c		2	20.e	even	4	2
7920.2.a.l		1	12.b	even	2	1
9680.2.a.d		1	44.c	even	2	1
9900.2.a.n		1	15.d	odd	2	1
9900.2.c.e		2	15.e	even	4	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{3} - 2$ T3 - 2 acting on $S_{2}^{\mathrm{new}}(\Gamma_0(220))$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T$ T
$3$	$T - 2$ T - 2
$5$	$T - 1$ T - 1
$7$	$T$ T
$11$	$T - 1$ T - 1
$13$	$T$ T
$17$	$T + 4$ T + 4
$19$	$T + 4$ T + 4
$23$	$T - 6$ T - 6
$29$	$T - 2$ T - 2
$31$	$T$ T
$37$	$T + 6$ T + 6
$41$	$T + 10$ T + 10
$43$	$T - 4$ T - 4
$47$	$T - 10$ T - 10
$53$	$T - 2$ T - 2
$59$	$T + 4$ T + 4
$61$	$T + 14$ T + 14
$67$	$T - 2$ T - 2
$71$	$T - 4$ T - 4
$73$	$T + 4$ T + 4
$79$	$T + 8$ T + 8
$83$	$T - 12$ T - 12
$89$	$T - 6$ T - 6
$97$	$T - 6$ T - 6
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