# Properties

 Label 2160.2.a.f Level $2160$ Weight $2$ Character orbit 2160.a Self dual yes Analytic conductor $17.248$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2160,2,Mod(1,2160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2160, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("2160.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2160 = 2^{4} \cdot 3^{3} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2160.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: $$17.2476868366$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1080) 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)

 $$f(q)$$ $$=$$ $$q - q^{5}+O(q^{10})$$ q - q^5 $$q - q^{5} + 2 q^{11} - 3 q^{17} + q^{19} + 3 q^{23} + q^{25} - 4 q^{29} + 5 q^{31} + 10 q^{37} - 6 q^{41} + 6 q^{43} + 8 q^{47} - 7 q^{49} - 3 q^{53} - 2 q^{55} + 5 q^{61} + 2 q^{67} + 2 q^{71} + 6 q^{73} + 11 q^{79} + 9 q^{83} + 3 q^{85} - 10 q^{89} - q^{95} + 8 q^{97}+O(q^{100})$$ q - q^5 + 2 * q^11 - 3 * q^17 + q^19 + 3 * q^23 + q^25 - 4 * q^29 + 5 * q^31 + 10 * q^37 - 6 * q^41 + 6 * q^43 + 8 * q^47 - 7 * q^49 - 3 * q^53 - 2 * q^55 + 5 * q^61 + 2 * q^67 + 2 * q^71 + 6 * q^73 + 11 * q^79 + 9 * q^83 + 3 * q^85 - 10 * q^89 - q^95 + 8 * q^97

## 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
 0
0 0 0 −1.00000 0 0 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$-1$$
$$5$$ $$1$$

## 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 2160.2.a.f 1
3.b odd 2 1 2160.2.a.r 1
4.b odd 2 1 1080.2.a.d 1
8.b even 2 1 8640.2.a.bs 1
8.d odd 2 1 8640.2.a.bt 1
12.b even 2 1 1080.2.a.j yes 1
20.d odd 2 1 5400.2.a.w 1
20.e even 4 2 5400.2.f.j 2
24.f even 2 1 8640.2.a.o 1
24.h odd 2 1 8640.2.a.p 1
36.f odd 6 2 3240.2.q.s 2
36.h even 6 2 3240.2.q.e 2
60.h even 2 1 5400.2.a.x 1
60.l odd 4 2 5400.2.f.r 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1080.2.a.d 1 4.b odd 2 1
1080.2.a.j yes 1 12.b even 2 1
2160.2.a.f 1 1.a even 1 1 trivial
2160.2.a.r 1 3.b odd 2 1
3240.2.q.e 2 36.h even 6 2
3240.2.q.s 2 36.f odd 6 2
5400.2.a.w 1 20.d odd 2 1
5400.2.a.x 1 60.h even 2 1
5400.2.f.j 2 20.e even 4 2
5400.2.f.r 2 60.l odd 4 2
8640.2.a.o 1 24.f even 2 1
8640.2.a.p 1 24.h odd 2 1
8640.2.a.bs 1 8.b even 2 1
8640.2.a.bt 1 8.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2160))$$:

 $$T_{7}$$ T7 $$T_{11} - 2$$ T11 - 2 $$T_{13}$$ T13 $$T_{17} + 3$$ T17 + 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T + 1$$
$7$ $$T$$
$11$ $$T - 2$$
$13$ $$T$$
$17$ $$T + 3$$
$19$ $$T - 1$$
$23$ $$T - 3$$
$29$ $$T + 4$$
$31$ $$T - 5$$
$37$ $$T - 10$$
$41$ $$T + 6$$
$43$ $$T - 6$$
$47$ $$T - 8$$
$53$ $$T + 3$$
$59$ $$T$$
$61$ $$T - 5$$
$67$ $$T - 2$$
$71$ $$T - 2$$
$73$ $$T - 6$$
$79$ $$T - 11$$
$83$ $$T - 9$$
$89$ $$T + 10$$
$97$ $$T - 8$$