# Properties

 Label 2160.2.a.n Level $2160$ Weight $2$ Character orbit 2160.a Self dual yes Analytic conductor $17.248$ Analytic rank $1$ 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: $$1$$ 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)

 $$f(q)$$ $$=$$ $$q + q^{5} - 2 q^{7}+O(q^{10})$$ q + q^5 - 2 * q^7 $$q + q^{5} - 2 q^{7} - q^{11} + q^{13} - q^{17} - 4 q^{19} - q^{23} + q^{25} - 5 q^{29} - q^{31} - 2 q^{35} + 6 q^{37} - 7 q^{43} - 7 q^{47} - 3 q^{49} - 12 q^{53} - q^{55} + 4 q^{59} + 10 q^{61} + q^{65} + 4 q^{67} - 12 q^{71} + 6 q^{73} + 2 q^{77} - 15 q^{79} - 2 q^{83} - q^{85} - 12 q^{89} - 2 q^{91} - 4 q^{95} + 10 q^{97}+O(q^{100})$$ q + q^5 - 2 * q^7 - q^11 + q^13 - q^17 - 4 * q^19 - q^23 + q^25 - 5 * q^29 - q^31 - 2 * q^35 + 6 * q^37 - 7 * q^43 - 7 * q^47 - 3 * q^49 - 12 * q^53 - q^55 + 4 * q^59 + 10 * q^61 + q^65 + 4 * q^67 - 12 * q^71 + 6 * q^73 + 2 * q^77 - 15 * q^79 - 2 * q^83 - q^85 - 12 * q^89 - 2 * q^91 - 4 * q^95 + 10 * 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 −2.00000 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.n 1
3.b odd 2 1 2160.2.a.d 1
4.b odd 2 1 1080.2.a.k yes 1
8.b even 2 1 8640.2.a.i 1
8.d odd 2 1 8640.2.a.v 1
12.b even 2 1 1080.2.a.f 1
20.d odd 2 1 5400.2.a.n 1
20.e even 4 2 5400.2.f.p 2
24.f even 2 1 8640.2.a.cb 1
24.h odd 2 1 8640.2.a.bk 1
36.f odd 6 2 3240.2.q.c 2
36.h even 6 2 3240.2.q.o 2
60.h even 2 1 5400.2.a.m 1
60.l odd 4 2 5400.2.f.m 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1080.2.a.f 1 12.b even 2 1
1080.2.a.k yes 1 4.b odd 2 1
2160.2.a.d 1 3.b odd 2 1
2160.2.a.n 1 1.a even 1 1 trivial
3240.2.q.c 2 36.f odd 6 2
3240.2.q.o 2 36.h even 6 2
5400.2.a.m 1 60.h even 2 1
5400.2.a.n 1 20.d odd 2 1
5400.2.f.m 2 60.l odd 4 2
5400.2.f.p 2 20.e even 4 2
8640.2.a.i 1 8.b even 2 1
8640.2.a.v 1 8.d odd 2 1
8640.2.a.bk 1 24.h odd 2 1
8640.2.a.cb 1 24.f even 2 1

## Hecke kernels

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

 $$T_{7} + 2$$ T7 + 2 $$T_{11} + 1$$ T11 + 1 $$T_{13} - 1$$ T13 - 1 $$T_{17} + 1$$ T17 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T - 1$$
$7$ $$T + 2$$
$11$ $$T + 1$$
$13$ $$T - 1$$
$17$ $$T + 1$$
$19$ $$T + 4$$
$23$ $$T + 1$$
$29$ $$T + 5$$
$31$ $$T + 1$$
$37$ $$T - 6$$
$41$ $$T$$
$43$ $$T + 7$$
$47$ $$T + 7$$
$53$ $$T + 12$$
$59$ $$T - 4$$
$61$ $$T - 10$$
$67$ $$T - 4$$
$71$ $$T + 12$$
$73$ $$T - 6$$
$79$ $$T + 15$$
$83$ $$T + 2$$
$89$ $$T + 12$$
$97$ $$T - 10$$