# Properties

 Label 2160.2.f.m 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: $$\Q(i, \sqrt{19})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 9x^{2} + 25$$ x^4 - 9*x^2 + 25 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 270) 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{5} + ( - 2 \beta_{3} - 1) q^{7}+O(q^{10})$$ q + b1 * q^5 + (-2*b3 - 1) * q^7 $$q + \beta_1 q^{5} + ( - 2 \beta_{3} - 1) q^{7} + (\beta_{2} - 2 \beta_1) q^{11} - 4 \beta_{2} q^{17} + 6 q^{19} + 2 \beta_{2} q^{23} + (\beta_{3} + 5) q^{25} - 7 q^{31} + ( - 10 \beta_{2} + \beta_1) q^{35} + ( - 4 \beta_{3} - 2) q^{37} + (2 \beta_{2} - 4 \beta_1) q^{41} + ( - 4 \beta_{3} - 2) q^{43} + 2 \beta_{2} q^{47} - 12 q^{49} - 3 \beta_{2} q^{53} + ( - \beta_{3} - 10) q^{55} + ( - 2 \beta_{2} + 4 \beta_1) q^{59} - 4 q^{61} + ( - 4 \beta_{3} - 2) q^{67} + ( - 2 \beta_{3} - 1) q^{73} + 19 \beta_{2} q^{77} + 5 \beta_{2} q^{83} - 4 \beta_{3} q^{85} + ( - 2 \beta_{2} + 4 \beta_1) q^{89} + 6 \beta_1 q^{95} + ( - 2 \beta_{3} - 1) q^{97}+O(q^{100})$$ q + b1 * q^5 + (-2*b3 - 1) * q^7 + (b2 - 2*b1) * q^11 - 4*b2 * q^17 + 6 * q^19 + 2*b2 * q^23 + (b3 + 5) * q^25 - 7 * q^31 + (-10*b2 + b1) * q^35 + (-4*b3 - 2) * q^37 + (2*b2 - 4*b1) * q^41 + (-4*b3 - 2) * q^43 + 2*b2 * q^47 - 12 * q^49 - 3*b2 * q^53 + (-b3 - 10) * q^55 + (-2*b2 + 4*b1) * q^59 - 4 * q^61 + (-4*b3 - 2) * q^67 + (-2*b3 - 1) * q^73 + 19*b2 * q^77 + 5*b2 * q^83 - 4*b3 * q^85 + (-2*b2 + 4*b1) * q^89 + 6*b1 * q^95 + (-2*b3 - 1) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q + 24 q^{19} + 18 q^{25} - 28 q^{31} - 48 q^{49} - 38 q^{55} - 16 q^{61} + 8 q^{85}+O(q^{100})$$ 4 * q + 24 * q^19 + 18 * q^25 - 28 * q^31 - 48 * q^49 - 38 * q^55 - 16 * q^61 + 8 * q^85

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 9x^{2} + 25$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{3} - 4\nu ) / 5$$ (v^3 - 4*v) / 5 $$\beta_{3}$$ $$=$$ $$\nu^{2} - 5$$ v^2 - 5
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} + 5$$ b3 + 5 $$\nu^{3}$$ $$=$$ $$5\beta_{2} + 4\beta_1$$ 5*b2 + 4*b1

## 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
 −2.17945 − 0.500000i −2.17945 + 0.500000i 2.17945 − 0.500000i 2.17945 + 0.500000i
0 0 0 −2.17945 0.500000i 0 4.35890i 0 0 0
1729.2 0 0 0 −2.17945 + 0.500000i 0 4.35890i 0 0 0
1729.3 0 0 0 2.17945 0.500000i 0 4.35890i 0 0 0
1729.4 0 0 0 2.17945 + 0.500000i 0 4.35890i 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.m 4
3.b odd 2 1 inner 2160.2.f.m 4
4.b odd 2 1 270.2.c.c 4
5.b even 2 1 inner 2160.2.f.m 4
12.b even 2 1 270.2.c.c 4
15.d odd 2 1 inner 2160.2.f.m 4
20.d odd 2 1 270.2.c.c 4
20.e even 4 1 1350.2.a.w 2
20.e even 4 1 1350.2.a.x 2
36.f odd 6 2 810.2.i.h 8
36.h even 6 2 810.2.i.h 8
60.h even 2 1 270.2.c.c 4
60.l odd 4 1 1350.2.a.w 2
60.l odd 4 1 1350.2.a.x 2
180.n even 6 2 810.2.i.h 8
180.p odd 6 2 810.2.i.h 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
270.2.c.c 4 4.b odd 2 1
270.2.c.c 4 12.b even 2 1
270.2.c.c 4 20.d odd 2 1
270.2.c.c 4 60.h even 2 1
810.2.i.h 8 36.f odd 6 2
810.2.i.h 8 36.h even 6 2
810.2.i.h 8 180.n even 6 2
810.2.i.h 8 180.p odd 6 2
1350.2.a.w 2 20.e even 4 1
1350.2.a.w 2 60.l odd 4 1
1350.2.a.x 2 20.e even 4 1
1350.2.a.x 2 60.l odd 4 1
2160.2.f.m 4 1.a even 1 1 trivial
2160.2.f.m 4 3.b odd 2 1 inner
2160.2.f.m 4 5.b even 2 1 inner
2160.2.f.m 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 $$S_{2}^{\mathrm{new}}(2160, [\chi])$$:

 $$T_{7}^{2} + 19$$ T7^2 + 19 $$T_{11}^{2} - 19$$ T11^2 - 19 $$T_{13}$$ T13

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4} - 9T^{2} + 25$$
$7$ $$(T^{2} + 19)^{2}$$
$11$ $$(T^{2} - 19)^{2}$$
$13$ $$T^{4}$$
$17$ $$(T^{2} + 16)^{2}$$
$19$ $$(T - 6)^{4}$$
$23$ $$(T^{2} + 4)^{2}$$
$29$ $$T^{4}$$
$31$ $$(T + 7)^{4}$$
$37$ $$(T^{2} + 76)^{2}$$
$41$ $$(T^{2} - 76)^{2}$$
$43$ $$(T^{2} + 76)^{2}$$
$47$ $$(T^{2} + 4)^{2}$$
$53$ $$(T^{2} + 9)^{2}$$
$59$ $$(T^{2} - 76)^{2}$$
$61$ $$(T + 4)^{4}$$
$67$ $$(T^{2} + 76)^{2}$$
$71$ $$T^{4}$$
$73$ $$(T^{2} + 19)^{2}$$
$79$ $$T^{4}$$
$83$ $$(T^{2} + 25)^{2}$$
$89$ $$(T^{2} - 76)^{2}$$
$97$ $$(T^{2} + 19)^{2}$$