# Properties

 Label 2160.1.c.b Level $2160$ Weight $1$ Character orbit 2160.c Self dual yes Analytic conductor $1.078$ Analytic rank $0$ Dimension $1$ Projective image $D_{3}$ CM discriminant -15 Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2160,1,Mod(1889,2160)]

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, 1, 1]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("2160.1889");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2160 = 2^{4} \cdot 3^{3} \cdot 5$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 2160.c (of order $$2$$, degree $$1$$, not minimal)

## 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: $$1.07798042729$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 135) Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.135.1 Artin image: $D_6$ Artin field: Galois closure of 6.0.1166400.1

## $q$-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 $$q + q^{5} - q^{17} + q^{19} + q^{23} + q^{25} + q^{31} - 2 q^{47} + q^{49} - q^{53} - q^{61} + q^{79} + q^{83} - q^{85} + q^{95}+O(q^{100})$$ q + q^5 - q^17 + q^19 + q^23 + q^25 + q^31 - 2 * q^47 + q^49 - q^53 - q^61 + q^79 + q^83 - q^85 + q^95

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2160\mathbb{Z}\right)^\times$$.

 $$n$$ $$271$$ $$1297$$ $$1621$$ $$2081$$ $$\chi(n)$$ $$0$$ $$1$$ $$0$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1889.1
 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

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.d odd 2 1 CM by $$\Q(\sqrt{-15})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2160.1.c.b 1
3.b odd 2 1 2160.1.c.a 1
4.b odd 2 1 135.1.d.a 1
5.b even 2 1 2160.1.c.a 1
12.b even 2 1 135.1.d.b yes 1
15.d odd 2 1 CM 2160.1.c.b 1
20.d odd 2 1 135.1.d.b yes 1
20.e even 4 2 675.1.c.c 2
36.f odd 6 2 405.1.h.b 2
36.h even 6 2 405.1.h.a 2
60.h even 2 1 135.1.d.a 1
60.l odd 4 2 675.1.c.c 2
108.j odd 18 6 3645.1.n.d 6
108.l even 18 6 3645.1.n.e 6
180.n even 6 2 405.1.h.b 2
180.p odd 6 2 405.1.h.a 2
180.v odd 12 4 2025.1.j.c 4
180.x even 12 4 2025.1.j.c 4
540.bb even 18 6 3645.1.n.d 6
540.bf odd 18 6 3645.1.n.e 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.1.d.a 1 4.b odd 2 1
135.1.d.a 1 60.h even 2 1
135.1.d.b yes 1 12.b even 2 1
135.1.d.b yes 1 20.d odd 2 1
405.1.h.a 2 36.h even 6 2
405.1.h.a 2 180.p odd 6 2
405.1.h.b 2 36.f odd 6 2
405.1.h.b 2 180.n even 6 2
675.1.c.c 2 20.e even 4 2
675.1.c.c 2 60.l odd 4 2
2025.1.j.c 4 180.v odd 12 4
2025.1.j.c 4 180.x even 12 4
2160.1.c.a 1 3.b odd 2 1
2160.1.c.a 1 5.b even 2 1
2160.1.c.b 1 1.a even 1 1 trivial
2160.1.c.b 1 15.d odd 2 1 CM
3645.1.n.d 6 108.j odd 18 6
3645.1.n.d 6 540.bb even 18 6
3645.1.n.e 6 108.l even 18 6
3645.1.n.e 6 540.bf odd 18 6

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(2160, [\chi])$$:

 $$T_{7}$$ T7 $$T_{17} + 1$$ T17 + 1

## Hecke characteristic polynomials

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