# Properties

 Label 2160.2.by.b Level $2160$ Weight $2$ Character orbit 2160.by Analytic conductor $17.248$ Analytic rank $0$ Dimension $4$ Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2160,2,Mod(289,2160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2160, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 4, 3]))

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

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

chi := DirichletCharacter("2160.289");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2160 = 2^{4} \cdot 3^{3} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2160.by (of order $$6$$, degree $$2$$, 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: no Analytic conductor: $$17.2476868366$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 90) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (2 \zeta_{12}^{3} + \cdots - 2 \zeta_{12}) q^{5}+ \cdots + \zeta_{12} q^{7}+O(q^{10})$$ q + (2*z^3 - z^2 - 2*z) * q^5 + z * q^7 $$q + (2 \zeta_{12}^{3} + \cdots - 2 \zeta_{12}) q^{5}+ \cdots - 8 \zeta_{12} q^{97} +O(q^{100})$$ q + (2*z^3 - z^2 - 2*z) * q^5 + z * q^7 + (-2*z^2 + 2) * q^11 + (6*z^3 - 6*z) * q^13 - 2*z^3 * q^17 + 6 * q^19 + (-z^3 + z) * q^23 + (-3*z^2 + 4*z + 3) * q^25 + (9*z^2 - 9) * q^29 - 2*z^2 * q^31 + (-z^3 - 2) * q^35 - 2*z^3 * q^37 - 11*z^2 * q^41 - 4*z * q^43 - 7*z * q^47 - 6*z^2 * q^49 + (4*z^3 - 2) * q^55 - 4*z^2 * q^59 + (-7*z^2 + 7) * q^61 + (-12*z^2 + 6*z + 12) * q^65 + (11*z^3 - 11*z) * q^67 - 6 * q^71 - 4*z^3 * q^73 + (-2*z^3 + 2*z) * q^77 + (-12*z^2 + 12) * q^79 + 11*z * q^83 + (2*z^3 + 4*z^2 - 2*z) * q^85 + q^89 - 6 * q^91 + (12*z^3 - 6*z^2 - 12*z) * q^95 - 8*z * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{5}+O(q^{10})$$ 4 * q - 2 * q^5 $$4 q - 2 q^{5} + 4 q^{11} + 24 q^{19} + 6 q^{25} - 18 q^{29} - 4 q^{31} - 8 q^{35} - 22 q^{41} - 12 q^{49} - 8 q^{55} - 8 q^{59} + 14 q^{61} + 24 q^{65} - 24 q^{71} + 24 q^{79} + 8 q^{85} + 4 q^{89} - 24 q^{91} - 12 q^{95}+O(q^{100})$$ 4 * q - 2 * q^5 + 4 * q^11 + 24 * q^19 + 6 * q^25 - 18 * q^29 - 4 * q^31 - 8 * q^35 - 22 * q^41 - 12 * q^49 - 8 * q^55 - 8 * q^59 + 14 * q^61 + 24 * q^65 - 24 * q^71 + 24 * q^79 + 8 * q^85 + 4 * q^89 - 24 * q^91 - 12 * q^95

## Character values

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

 $$n$$ $$271$$ $$1297$$ $$1621$$ $$2081$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$ $$-\zeta_{12}^{2}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
289.1
 0.866025 + 0.500000i −0.866025 − 0.500000i 0.866025 − 0.500000i −0.866025 + 0.500000i
0 0 0 −2.23205 + 0.133975i 0 0.866025 + 0.500000i 0 0 0
289.2 0 0 0 1.23205 1.86603i 0 −0.866025 0.500000i 0 0 0
1009.1 0 0 0 −2.23205 0.133975i 0 0.866025 0.500000i 0 0 0
1009.2 0 0 0 1.23205 + 1.86603i 0 −0.866025 + 0.500000i 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
5.b even 2 1 inner
9.c even 3 1 inner
45.j even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2160.2.by.b 4
3.b odd 2 1 720.2.by.a 4
4.b odd 2 1 270.2.i.a 4
5.b even 2 1 inner 2160.2.by.b 4
9.c even 3 1 inner 2160.2.by.b 4
9.d odd 6 1 720.2.by.a 4
12.b even 2 1 90.2.i.a 4
15.d odd 2 1 720.2.by.a 4
20.d odd 2 1 270.2.i.a 4
20.e even 4 1 1350.2.e.a 2
20.e even 4 1 1350.2.e.i 2
36.f odd 6 1 270.2.i.a 4
36.f odd 6 1 810.2.c.c 2
36.h even 6 1 90.2.i.a 4
36.h even 6 1 810.2.c.b 2
45.h odd 6 1 720.2.by.a 4
45.j even 6 1 inner 2160.2.by.b 4
60.h even 2 1 90.2.i.a 4
60.l odd 4 1 450.2.e.b 2
60.l odd 4 1 450.2.e.g 2
180.n even 6 1 90.2.i.a 4
180.n even 6 1 810.2.c.b 2
180.p odd 6 1 270.2.i.a 4
180.p odd 6 1 810.2.c.c 2
180.v odd 12 1 450.2.e.b 2
180.v odd 12 1 450.2.e.g 2
180.v odd 12 1 4050.2.a.j 1
180.v odd 12 1 4050.2.a.x 1
180.x even 12 1 1350.2.e.a 2
180.x even 12 1 1350.2.e.i 2
180.x even 12 1 4050.2.a.g 1
180.x even 12 1 4050.2.a.be 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.2.i.a 4 12.b even 2 1
90.2.i.a 4 36.h even 6 1
90.2.i.a 4 60.h even 2 1
90.2.i.a 4 180.n even 6 1
270.2.i.a 4 4.b odd 2 1
270.2.i.a 4 20.d odd 2 1
270.2.i.a 4 36.f odd 6 1
270.2.i.a 4 180.p odd 6 1
450.2.e.b 2 60.l odd 4 1
450.2.e.b 2 180.v odd 12 1
450.2.e.g 2 60.l odd 4 1
450.2.e.g 2 180.v odd 12 1
720.2.by.a 4 3.b odd 2 1
720.2.by.a 4 9.d odd 6 1
720.2.by.a 4 15.d odd 2 1
720.2.by.a 4 45.h odd 6 1
810.2.c.b 2 36.h even 6 1
810.2.c.b 2 180.n even 6 1
810.2.c.c 2 36.f odd 6 1
810.2.c.c 2 180.p odd 6 1
1350.2.e.a 2 20.e even 4 1
1350.2.e.a 2 180.x even 12 1
1350.2.e.i 2 20.e even 4 1
1350.2.e.i 2 180.x even 12 1
2160.2.by.b 4 1.a even 1 1 trivial
2160.2.by.b 4 5.b even 2 1 inner
2160.2.by.b 4 9.c even 3 1 inner
2160.2.by.b 4 45.j even 6 1 inner
4050.2.a.g 1 180.x even 12 1
4050.2.a.j 1 180.v odd 12 1
4050.2.a.x 1 180.v odd 12 1
4050.2.a.be 1 180.x even 12 1

## Hecke kernels

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

 $$T_{7}^{4} - T_{7}^{2} + 1$$ T7^4 - T7^2 + 1 $$T_{11}^{2} - 2T_{11} + 4$$ T11^2 - 2*T11 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4} + 2 T^{3} + \cdots + 25$$
$7$ $$T^{4} - T^{2} + 1$$
$11$ $$(T^{2} - 2 T + 4)^{2}$$
$13$ $$T^{4} - 36T^{2} + 1296$$
$17$ $$(T^{2} + 4)^{2}$$
$19$ $$(T - 6)^{4}$$
$23$ $$T^{4} - T^{2} + 1$$
$29$ $$(T^{2} + 9 T + 81)^{2}$$
$31$ $$(T^{2} + 2 T + 4)^{2}$$
$37$ $$(T^{2} + 4)^{2}$$
$41$ $$(T^{2} + 11 T + 121)^{2}$$
$43$ $$T^{4} - 16T^{2} + 256$$
$47$ $$T^{4} - 49T^{2} + 2401$$
$53$ $$T^{4}$$
$59$ $$(T^{2} + 4 T + 16)^{2}$$
$61$ $$(T^{2} - 7 T + 49)^{2}$$
$67$ $$T^{4} - 121 T^{2} + 14641$$
$71$ $$(T + 6)^{4}$$
$73$ $$(T^{2} + 16)^{2}$$
$79$ $$(T^{2} - 12 T + 144)^{2}$$
$83$ $$T^{4} - 121 T^{2} + 14641$$
$89$ $$(T - 1)^{4}$$
$97$ $$T^{4} - 64T^{2} + 4096$$