# Properties

 Label 2160.2.f.k Level $2160$ Weight $2$ Character orbit 2160.f 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(1729,2160)]

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, 1]))

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

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

chi := DirichletCharacter("2160.1729");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2160 = 2^{4} \cdot 3^{3} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2160.f (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: no Analytic conductor: $$17.2476868366$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$3^{2}$$ Twist minimal: no (minimal twist has level 135) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{5} + \beta_1 q^{7}+O(q^{10})$$ q + b3 * q^5 + b1 * q^7 $$q + \beta_{3} q^{5} + \beta_1 q^{7} + (2 \beta_{3} - \beta_{2}) q^{11} + \beta_1 q^{13} - 2 \beta_{2} q^{17} + q^{19} + 5 \beta_{2} q^{23} + (\beta_1 + 4) q^{25} + ( - 2 \beta_{3} + \beta_{2}) q^{29} - 2 q^{31} + ( - \beta_{3} + 5 \beta_{2}) q^{35} - 3 \beta_1 q^{37} + (2 \beta_{3} - \beta_{2}) q^{41} - 2 \beta_1 q^{43} + 2 \beta_{2} q^{47} - 2 q^{49} + 7 \beta_{2} q^{53} + (\beta_1 + 9) q^{55} + (4 \beta_{3} - 2 \beta_{2}) q^{59} - 13 q^{61} + ( - \beta_{3} + 5 \beta_{2}) q^{65} - \beta_1 q^{67} + ( - 6 \beta_{3} + 3 \beta_{2}) q^{71} + 3 \beta_1 q^{73} + 9 \beta_{2} q^{77} - 5 q^{79} - \beta_{2} q^{83} + ( - 2 \beta_1 + 2) q^{85} - 9 q^{91} + \beta_{3} q^{95} - \beta_1 q^{97}+O(q^{100})$$ q + b3 * q^5 + b1 * q^7 + (2*b3 - b2) * q^11 + b1 * q^13 - 2*b2 * q^17 + q^19 + 5*b2 * q^23 + (b1 + 4) * q^25 + (-2*b3 + b2) * q^29 - 2 * q^31 + (-b3 + 5*b2) * q^35 - 3*b1 * q^37 + (2*b3 - b2) * q^41 - 2*b1 * q^43 + 2*b2 * q^47 - 2 * q^49 + 7*b2 * q^53 + (b1 + 9) * q^55 + (4*b3 - 2*b2) * q^59 - 13 * q^61 + (-b3 + 5*b2) * q^65 - b1 * q^67 + (-6*b3 + 3*b2) * q^71 + 3*b1 * q^73 + 9*b2 * q^77 - 5 * q^79 - b2 * q^83 + (-2*b1 + 2) * q^85 - 9 * q^91 + b3 * q^95 - b1 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q + 4 q^{19} + 16 q^{25} - 8 q^{31} - 8 q^{49} + 36 q^{55} - 52 q^{61} - 20 q^{79} + 8 q^{85} - 36 q^{91}+O(q^{100})$$ 4 * q + 4 * q^19 + 16 * q^25 - 8 * q^31 - 8 * q^49 + 36 * q^55 - 52 * q^61 - 20 * q^79 + 8 * q^85 - 36 * q^91

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$3\zeta_{8}^{2}$$ 3*v^2 $$\beta_{2}$$ $$=$$ $$\zeta_{8}^{3} + \zeta_{8}$$ v^3 + v $$\beta_{3}$$ $$=$$ $$-\zeta_{8}^{3} + 2\zeta_{8}$$ -v^3 + 2*v
 $$\zeta_{8}$$ $$=$$ $$( \beta_{3} + \beta_{2} ) / 3$$ (b3 + b2) / 3 $$\zeta_{8}^{2}$$ $$=$$ $$( \beta_1 ) / 3$$ (b1) / 3 $$\zeta_{8}^{3}$$ $$=$$ $$( -\beta_{3} + 2\beta_{2} ) / 3$$ (-b3 + 2*b2) / 3

## Character values

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

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

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1729.1
 −0.707107 − 0.707107i −0.707107 + 0.707107i 0.707107 − 0.707107i 0.707107 + 0.707107i
0 0 0 −2.12132 0.707107i 0 3.00000i 0 0 0
1729.2 0 0 0 −2.12132 + 0.707107i 0 3.00000i 0 0 0
1729.3 0 0 0 2.12132 0.707107i 0 3.00000i 0 0 0
1729.4 0 0 0 2.12132 + 0.707107i 0 3.00000i 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
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2160.2.f.k 4
3.b odd 2 1 inner 2160.2.f.k 4
4.b odd 2 1 135.2.b.b 4
5.b even 2 1 inner 2160.2.f.k 4
12.b even 2 1 135.2.b.b 4
15.d odd 2 1 inner 2160.2.f.k 4
20.d odd 2 1 135.2.b.b 4
20.e even 4 1 675.2.a.l 2
20.e even 4 1 675.2.a.m 2
36.f odd 6 2 405.2.j.f 8
36.h even 6 2 405.2.j.f 8
60.h even 2 1 135.2.b.b 4
60.l odd 4 1 675.2.a.l 2
60.l odd 4 1 675.2.a.m 2
180.n even 6 2 405.2.j.f 8
180.p odd 6 2 405.2.j.f 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.2.b.b 4 4.b odd 2 1
135.2.b.b 4 12.b even 2 1
135.2.b.b 4 20.d odd 2 1
135.2.b.b 4 60.h even 2 1
405.2.j.f 8 36.f odd 6 2
405.2.j.f 8 36.h even 6 2
405.2.j.f 8 180.n even 6 2
405.2.j.f 8 180.p odd 6 2
675.2.a.l 2 20.e even 4 1
675.2.a.l 2 60.l odd 4 1
675.2.a.m 2 20.e even 4 1
675.2.a.m 2 60.l odd 4 1
2160.2.f.k 4 1.a even 1 1 trivial
2160.2.f.k 4 3.b odd 2 1 inner
2160.2.f.k 4 5.b even 2 1 inner
2160.2.f.k 4 15.d odd 2 1 inner

## Hecke kernels

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

 $$T_{7}^{2} + 9$$ T7^2 + 9 $$T_{11}^{2} - 18$$ T11^2 - 18 $$T_{13}^{2} + 9$$ T13^2 + 9

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4} - 8T^{2} + 25$$
$7$ $$(T^{2} + 9)^{2}$$
$11$ $$(T^{2} - 18)^{2}$$
$13$ $$(T^{2} + 9)^{2}$$
$17$ $$(T^{2} + 8)^{2}$$
$19$ $$(T - 1)^{4}$$
$23$ $$(T^{2} + 50)^{2}$$
$29$ $$(T^{2} - 18)^{2}$$
$31$ $$(T + 2)^{4}$$
$37$ $$(T^{2} + 81)^{2}$$
$41$ $$(T^{2} - 18)^{2}$$
$43$ $$(T^{2} + 36)^{2}$$
$47$ $$(T^{2} + 8)^{2}$$
$53$ $$(T^{2} + 98)^{2}$$
$59$ $$(T^{2} - 72)^{2}$$
$61$ $$(T + 13)^{4}$$
$67$ $$(T^{2} + 9)^{2}$$
$71$ $$(T^{2} - 162)^{2}$$
$73$ $$(T^{2} + 81)^{2}$$
$79$ $$(T + 5)^{4}$$
$83$ $$(T^{2} + 2)^{2}$$
$89$ $$T^{4}$$
$97$ $$(T^{2} + 9)^{2}$$