# Properties

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

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2160,2,Mod(721,2160)]

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, 0]))

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

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

chi := DirichletCharacter("2160.721");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2160 = 2^{4} \cdot 3^{3} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2160.q (of order $$3$$, 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 $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-11})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 2x^{2} - 3x + 9$$ x^4 - x^3 - 2*x^2 - 3*x + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3$$ Twist minimal: no (minimal twist has level 90) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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 basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_1 - 1) q^{5} + \beta_{3} q^{7}+O(q^{10})$$ q + (b1 - 1) * q^5 + b3 * q^7 $$q + (\beta_1 - 1) q^{5} + \beta_{3} q^{7} + (\beta_{3} + \beta_1) q^{11} + ( - 2 \beta_{3} + 2 \beta_{2} + 2) q^{13} + ( - \beta_{2} + 4) q^{17} + \beta_{2} q^{19} + (\beta_{3} - \beta_{2} - 2 \beta_1 + 1) q^{23} - \beta_1 q^{25} + (\beta_{3} - 2 \beta_1) q^{29} + ( - 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{31} + ( - \beta_{2} - 1) q^{35} - 4 q^{37} + (3 \beta_1 - 3) q^{41} + (\beta_{3} - 9 \beta_1) q^{43} + (\beta_{3} + 4 \beta_1) q^{47} + (\beta_{3} - \beta_{2} + \beta_1 - 2) q^{49} + (4 \beta_{2} + 2) q^{53} + ( - \beta_{2} - 2) q^{55} + (\beta_{3} - \beta_{2} + \beta_1 - 2) q^{59} + (3 \beta_{3} - 2 \beta_1) q^{61} + 2 \beta_{3} q^{65} + (7 \beta_1 - 7) q^{67} - 6 q^{71} + (3 \beta_{2} - 4) q^{73} + (2 \beta_{3} - 2 \beta_{2} + 8 \beta_1 - 10) q^{77} + 2 \beta_1 q^{79} + ( - \beta_{3} - 4 \beta_1) q^{83} + ( - \beta_{3} + \beta_{2} + 5 \beta_1 - 4) q^{85} + ( - 3 \beta_{2} - 9) q^{89} + (2 \beta_{2} + 18) q^{91} + (\beta_{3} - \beta_{2} - \beta_1) q^{95} + (\beta_{3} + 5 \beta_1) q^{97}+O(q^{100})$$ q + (b1 - 1) * q^5 + b3 * q^7 + (b3 + b1) * q^11 + (-2*b3 + 2*b2 + 2) * q^13 + (-b2 + 4) * q^17 + b2 * q^19 + (b3 - b2 - 2*b1 + 1) * q^23 - b1 * q^25 + (b3 - 2*b1) * q^29 + (-2*b3 + 2*b2 + 2*b1) * q^31 + (-b2 - 1) * q^35 - 4 * q^37 + (3*b1 - 3) * q^41 + (b3 - 9*b1) * q^43 + (b3 + 4*b1) * q^47 + (b3 - b2 + b1 - 2) * q^49 + (4*b2 + 2) * q^53 + (-b2 - 2) * q^55 + (b3 - b2 + b1 - 2) * q^59 + (3*b3 - 2*b1) * q^61 + 2*b3 * q^65 + (7*b1 - 7) * q^67 - 6 * q^71 + (3*b2 - 4) * q^73 + (2*b3 - 2*b2 + 8*b1 - 10) * q^77 + 2*b1 * q^79 + (-b3 - 4*b1) * q^83 + (-b3 + b2 + 5*b1 - 4) * q^85 + (-3*b2 - 9) * q^89 + (2*b2 + 18) * q^91 + (b3 - b2 - b1) * q^95 + (b3 + 5*b1) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{5} + q^{7}+O(q^{10})$$ 4 * q - 2 * q^5 + q^7 $$4 q - 2 q^{5} + q^{7} + 3 q^{11} + 2 q^{13} + 18 q^{17} - 2 q^{19} + 3 q^{23} - 2 q^{25} - 3 q^{29} - 2 q^{31} - 2 q^{35} - 16 q^{37} - 6 q^{41} - 17 q^{43} + 9 q^{47} - 3 q^{49} - 6 q^{55} - 3 q^{59} - q^{61} + 2 q^{65} - 14 q^{67} - 24 q^{71} - 22 q^{73} - 18 q^{77} + 4 q^{79} - 9 q^{83} - 9 q^{85} - 30 q^{89} + 68 q^{91} + q^{95} + 11 q^{97}+O(q^{100})$$ 4 * q - 2 * q^5 + q^7 + 3 * q^11 + 2 * q^13 + 18 * q^17 - 2 * q^19 + 3 * q^23 - 2 * q^25 - 3 * q^29 - 2 * q^31 - 2 * q^35 - 16 * q^37 - 6 * q^41 - 17 * q^43 + 9 * q^47 - 3 * q^49 - 6 * q^55 - 3 * q^59 - q^61 + 2 * q^65 - 14 * q^67 - 24 * q^71 - 22 * q^73 - 18 * q^77 + 4 * q^79 - 9 * q^83 - 9 * q^85 - 30 * q^89 + 68 * q^91 + q^95 + 11 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 2x^{2} - 3x + 9$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} + 2\nu^{2} - 2\nu - 3 ) / 6$$ (v^3 + 2*v^2 - 2*v - 3) / 6 $$\beta_{2}$$ $$=$$ $$( -\nu^{3} + \nu^{2} + 5\nu ) / 3$$ (-v^3 + v^2 + 5*v) / 3 $$\beta_{3}$$ $$=$$ $$( 2\nu^{3} + \nu^{2} + 2\nu - 9 ) / 3$$ (2*v^3 + v^2 + 2*v - 9) / 3
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_{2} - 2\beta _1 + 2 ) / 3$$ (b3 + b2 - 2*b1 + 2) / 3 $$\nu^{2}$$ $$=$$ $$( -\beta_{3} + 2\beta_{2} + 8\beta _1 + 1 ) / 3$$ (-b3 + 2*b2 + 8*b1 + 1) / 3 $$\nu^{3}$$ $$=$$ $$( 4\beta_{3} - 2\beta_{2} - 2\beta _1 + 11 ) / 3$$ (4*b3 - 2*b2 - 2*b1 + 11) / 3

## Character values

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

 $$n$$ $$271$$ $$1297$$ $$1621$$ $$2081$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-1 + \beta_{1}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
721.1
 −1.18614 + 1.26217i 1.68614 − 0.396143i −1.18614 − 1.26217i 1.68614 + 0.396143i
0 0 0 −0.500000 0.866025i 0 −1.18614 + 2.05446i 0 0 0
721.2 0 0 0 −0.500000 0.866025i 0 1.68614 2.92048i 0 0 0
1441.1 0 0 0 −0.500000 + 0.866025i 0 −1.18614 2.05446i 0 0 0
1441.2 0 0 0 −0.500000 + 0.866025i 0 1.68614 + 2.92048i 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
9.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2160.2.q.f 4
3.b odd 2 1 720.2.q.f 4
4.b odd 2 1 270.2.e.c 4
9.c even 3 1 inner 2160.2.q.f 4
9.c even 3 1 6480.2.a.bn 2
9.d odd 6 1 720.2.q.f 4
9.d odd 6 1 6480.2.a.be 2
12.b even 2 1 90.2.e.c 4
20.d odd 2 1 1350.2.e.l 4
20.e even 4 2 1350.2.j.f 8
36.f odd 6 1 270.2.e.c 4
36.f odd 6 1 810.2.a.k 2
36.h even 6 1 90.2.e.c 4
36.h even 6 1 810.2.a.i 2
60.h even 2 1 450.2.e.j 4
60.l odd 4 2 450.2.j.g 8
180.n even 6 1 450.2.e.j 4
180.n even 6 1 4050.2.a.bw 2
180.p odd 6 1 1350.2.e.l 4
180.p odd 6 1 4050.2.a.bo 2
180.v odd 12 2 450.2.j.g 8
180.v odd 12 2 4050.2.c.v 4
180.x even 12 2 1350.2.j.f 8
180.x even 12 2 4050.2.c.ba 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.2.e.c 4 12.b even 2 1
90.2.e.c 4 36.h even 6 1
270.2.e.c 4 4.b odd 2 1
270.2.e.c 4 36.f odd 6 1
450.2.e.j 4 60.h even 2 1
450.2.e.j 4 180.n even 6 1
450.2.j.g 8 60.l odd 4 2
450.2.j.g 8 180.v odd 12 2
720.2.q.f 4 3.b odd 2 1
720.2.q.f 4 9.d odd 6 1
810.2.a.i 2 36.h even 6 1
810.2.a.k 2 36.f odd 6 1
1350.2.e.l 4 20.d odd 2 1
1350.2.e.l 4 180.p odd 6 1
1350.2.j.f 8 20.e even 4 2
1350.2.j.f 8 180.x even 12 2
2160.2.q.f 4 1.a even 1 1 trivial
2160.2.q.f 4 9.c even 3 1 inner
4050.2.a.bo 2 180.p odd 6 1
4050.2.a.bw 2 180.n even 6 1
4050.2.c.v 4 180.v odd 12 2
4050.2.c.ba 4 180.x even 12 2
6480.2.a.be 2 9.d odd 6 1
6480.2.a.bn 2 9.c even 3 1

## Hecke kernels

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

 $$T_{7}^{4} - T_{7}^{3} + 9T_{7}^{2} + 8T_{7} + 64$$ T7^4 - T7^3 + 9*T7^2 + 8*T7 + 64 $$T_{11}^{4} - 3T_{11}^{3} + 15T_{11}^{2} + 18T_{11} + 36$$ T11^4 - 3*T11^3 + 15*T11^2 + 18*T11 + 36

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$(T^{2} + T + 1)^{2}$$
$7$ $$T^{4} - T^{3} + \cdots + 64$$
$11$ $$T^{4} - 3 T^{3} + \cdots + 36$$
$13$ $$T^{4} - 2 T^{3} + \cdots + 1024$$
$17$ $$(T^{2} - 9 T + 12)^{2}$$
$19$ $$(T^{2} + T - 8)^{2}$$
$23$ $$T^{4} - 3 T^{3} + \cdots + 36$$
$29$ $$T^{4} + 3 T^{3} + \cdots + 36$$
$31$ $$T^{4} + 2 T^{3} + \cdots + 1024$$
$37$ $$(T + 4)^{4}$$
$41$ $$(T^{2} + 3 T + 9)^{2}$$
$43$ $$T^{4} + 17 T^{3} + \cdots + 4096$$
$47$ $$T^{4} - 9 T^{3} + \cdots + 144$$
$53$ $$(T^{2} - 132)^{2}$$
$59$ $$T^{4} + 3 T^{3} + \cdots + 36$$
$61$ $$T^{4} + T^{3} + \cdots + 5476$$
$67$ $$(T^{2} + 7 T + 49)^{2}$$
$71$ $$(T + 6)^{4}$$
$73$ $$(T^{2} + 11 T - 44)^{2}$$
$79$ $$(T^{2} - 2 T + 4)^{2}$$
$83$ $$T^{4} + 9 T^{3} + \cdots + 144$$
$89$ $$(T^{2} + 15 T - 18)^{2}$$
$97$ $$T^{4} - 11 T^{3} + \cdots + 484$$