# Properties

 Label 2160.2.f.j Level $2160$ Weight $2$ Character orbit 2160.f Analytic conductor $17.248$ Analytic rank $0$ Dimension $4$ CM discriminant -15 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: $$\Q(i, \sqrt{5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 3x^{2} + 1$$ x^4 + 3*x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{2}\cdot 3^{2}$$ Twist minimal: no (minimal twist has level 135) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{5}+O(q^{10})$$ q + b1 * q^5 $$q + \beta_1 q^{5} + ( - 2 \beta_{2} - \beta_1) q^{17} + ( - \beta_{3} - 2) q^{19} + ( - \beta_{2} - 2 \beta_1) q^{23} - 5 q^{25} + ( - \beta_{3} + 4) q^{31} + 4 \beta_1 q^{47} + 7 q^{49} + (4 \beta_{2} - \beta_1) q^{53} + ( - 2 \beta_{3} - 1) q^{61} + ( - \beta_{3} - 8) q^{79} + ( - \beta_{2} + 4 \beta_1) q^{83} + ( - 2 \beta_{3} + 5) q^{85} + (5 \beta_{2} - 2 \beta_1) q^{95}+O(q^{100})$$ q + b1 * q^5 + (-2*b2 - b1) * q^17 + (-b3 - 2) * q^19 + (-b2 - 2*b1) * q^23 - 5 * q^25 + (-b3 + 4) * q^31 + 4*b1 * q^47 + 7 * q^49 + (4*b2 - b1) * q^53 + (-2*b3 - 1) * q^61 + (-b3 - 8) * q^79 + (-b2 + 4*b1) * q^83 + (-2*b3 + 5) * q^85 + (5*b2 - 2*b1) * q^95 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q - 8 q^{19} - 20 q^{25} + 16 q^{31} + 28 q^{49} - 4 q^{61} - 32 q^{79} + 20 q^{85}+O(q^{100})$$ 4 * q - 8 * q^19 - 20 * q^25 + 16 * q^31 + 28 * q^49 - 4 * q^61 - 32 * q^79 + 20 * q^85

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 3x^{2} + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu^{3} + 4\nu$$ v^3 + 4*v $$\beta_{2}$$ $$=$$ $$-3\nu^{3} - 6\nu$$ -3*v^3 - 6*v $$\beta_{3}$$ $$=$$ $$6\nu^{2} + 9$$ 6*v^2 + 9
 $$\nu$$ $$=$$ $$( \beta_{2} + 3\beta_1 ) / 6$$ (b2 + 3*b1) / 6 $$\nu^{2}$$ $$=$$ $$( \beta_{3} - 9 ) / 6$$ (b3 - 9) / 6 $$\nu^{3}$$ $$=$$ $$( -2\beta_{2} - 3\beta_1 ) / 3$$ (-2*b2 - 3*b1) / 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
 − 0.618034i − 1.61803i 1.61803i 0.618034i
0 0 0 2.23607i 0 0 0 0 0
1729.2 0 0 0 2.23607i 0 0 0 0 0
1729.3 0 0 0 2.23607i 0 0 0 0 0
1729.4 0 0 0 2.23607i 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 $$\Q(\sqrt{-15})$$
3.b odd 2 1 inner
5.b even 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.j 4
3.b odd 2 1 inner 2160.2.f.j 4
4.b odd 2 1 135.2.b.a 4
5.b even 2 1 inner 2160.2.f.j 4
12.b even 2 1 135.2.b.a 4
15.d odd 2 1 CM 2160.2.f.j 4
20.d odd 2 1 135.2.b.a 4
20.e even 4 1 675.2.a.j 2
20.e even 4 1 675.2.a.q 2
36.f odd 6 2 405.2.j.h 8
36.h even 6 2 405.2.j.h 8
60.h even 2 1 135.2.b.a 4
60.l odd 4 1 675.2.a.j 2
60.l odd 4 1 675.2.a.q 2
180.n even 6 2 405.2.j.h 8
180.p odd 6 2 405.2.j.h 8

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

## Hecke kernels

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

 $$T_{7}$$ T7 $$T_{11}$$ T11 $$T_{13}$$ T13

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$(T^{2} + 5)^{2}$$
$7$ $$T^{4}$$
$11$ $$T^{4}$$
$13$ $$T^{4}$$
$17$ $$T^{4} + 82T^{2} + 961$$
$19$ $$(T^{2} + 4 T - 41)^{2}$$
$23$ $$T^{4} + 58T^{2} + 121$$
$29$ $$T^{4}$$
$31$ $$(T^{2} - 8 T - 29)^{2}$$
$37$ $$T^{4}$$
$41$ $$T^{4}$$
$43$ $$T^{4}$$
$47$ $$(T^{2} + 80)^{2}$$
$53$ $$T^{4} + 298 T^{2} + 19321$$
$59$ $$T^{4}$$
$61$ $$(T^{2} + 2 T - 179)^{2}$$
$67$ $$T^{4}$$
$71$ $$T^{4}$$
$73$ $$T^{4}$$
$79$ $$(T^{2} + 16 T + 19)^{2}$$
$83$ $$T^{4} + 178T^{2} + 5041$$
$89$ $$T^{4}$$
$97$ $$T^{4}$$