# Properties

 Label 2160.2.f.i Level $2160$ Weight $2$ Character orbit 2160.f 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(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: $$\Q(i, \sqrt{6})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 9$$ x^4 + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1080) 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.

 $$f(q)$$ $$=$$ $$q + (\beta_{2} - \beta_1 - 1) q^{5} - \beta_{2} q^{7}+O(q^{10})$$ q + (b2 - b1 - 1) * q^5 - b2 * q^7 $$q + (\beta_{2} - \beta_1 - 1) q^{5} - \beta_{2} q^{7} + ( - \beta_{3} + \beta_1 - 2) q^{11} + ( - 2 \beta_{3} + \beta_{2} - 2 \beta_1) q^{13} + (2 \beta_{3} + 2 \beta_1) q^{17} + (2 \beta_{3} - 2 \beta_1 - 1) q^{19} + (\beta_{3} + 2 \beta_{2} + \beta_1) q^{23} + ( - 2 \beta_{3} + \beta_{2} + 2 \beta_1) q^{25} + ( - \beta_{3} + \beta_1 - 2) q^{29} - 6 q^{31} + (\beta_{3} + \beta_{2} + 1) q^{35} + ( - 2 \beta_{3} + 5 \beta_{2} - 2 \beta_1) q^{37} + ( - 3 \beta_{3} + 3 \beta_1 + 2) q^{41} - 6 \beta_{2} q^{43} + (2 \beta_{3} + 2 \beta_1) q^{47} + 6 q^{49} + (\beta_{3} + 2 \beta_{2} + \beta_1) q^{53} + (2 \beta_{3} - 5 \beta_{2} + 2 \beta_1 - 1) q^{55} + ( - 2 \beta_{3} + 2 \beta_1) q^{59} + ( - 4 \beta_{3} + 4 \beta_1 - 1) q^{61} + ( - \beta_{3} + 5 \beta_{2} + 4 \beta_1 - 7) q^{65} + (4 \beta_{3} + 5 \beta_{2} + 4 \beta_1) q^{67} + ( - \beta_{3} + \beta_1 - 14) q^{71} + (2 \beta_{3} - \beta_{2} + 2 \beta_1) q^{73} + ( - \beta_{3} + 2 \beta_{2} - \beta_1) q^{77} + (2 \beta_{3} - 2 \beta_1 + 1) q^{79} + ( - \beta_{3} + 10 \beta_{2} - \beta_1) q^{83} + ( - 6 \beta_{2} - 4 \beta_1 + 6) q^{85} - 12 q^{89} + (2 \beta_{3} - 2 \beta_1 + 1) q^{91} + ( - 4 \beta_{3} + 5 \beta_{2} + \cdots + 7) q^{95}+ \cdots + (2 \beta_{3} + 11 \beta_{2} + 2 \beta_1) q^{97}+O(q^{100})$$ q + (b2 - b1 - 1) * q^5 - b2 * q^7 + (-b3 + b1 - 2) * q^11 + (-2*b3 + b2 - 2*b1) * q^13 + (2*b3 + 2*b1) * q^17 + (2*b3 - 2*b1 - 1) * q^19 + (b3 + 2*b2 + b1) * q^23 + (-2*b3 + b2 + 2*b1) * q^25 + (-b3 + b1 - 2) * q^29 - 6 * q^31 + (b3 + b2 + 1) * q^35 + (-2*b3 + 5*b2 - 2*b1) * q^37 + (-3*b3 + 3*b1 + 2) * q^41 - 6*b2 * q^43 + (2*b3 + 2*b1) * q^47 + 6 * q^49 + (b3 + 2*b2 + b1) * q^53 + (2*b3 - 5*b2 + 2*b1 - 1) * q^55 + (-2*b3 + 2*b1) * q^59 + (-4*b3 + 4*b1 - 1) * q^61 + (-b3 + 5*b2 + 4*b1 - 7) * q^65 + (4*b3 + 5*b2 + 4*b1) * q^67 + (-b3 + b1 - 14) * q^71 + (2*b3 - b2 + 2*b1) * q^73 + (-b3 + 2*b2 - b1) * q^77 + (2*b3 - 2*b1 + 1) * q^79 + (-b3 + 10*b2 - b1) * q^83 + (-6*b2 - 4*b1 + 6) * q^85 - 12 * q^89 + (2*b3 - 2*b1 + 1) * q^91 + (-4*b3 + 5*b2 + b1 + 7) * q^95 + (2*b3 + 11*b2 + 2*b1) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{5}+O(q^{10})$$ 4 * q - 4 * q^5 $$4 q - 4 q^{5} - 8 q^{11} - 4 q^{19} - 8 q^{29} - 24 q^{31} + 4 q^{35} + 8 q^{41} + 24 q^{49} - 4 q^{55} - 4 q^{61} - 28 q^{65} - 56 q^{71} + 4 q^{79} + 24 q^{85} - 48 q^{89} + 4 q^{91} + 28 q^{95}+O(q^{100})$$ 4 * q - 4 * q^5 - 8 * q^11 - 4 * q^19 - 8 * q^29 - 24 * q^31 + 4 * q^35 + 8 * q^41 + 24 * q^49 - 4 * q^55 - 4 * q^61 - 28 * q^65 - 56 * q^71 + 4 * q^79 + 24 * q^85 - 48 * q^89 + 4 * q^91 + 28 * q^95

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

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

## 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
 1.22474 + 1.22474i 1.22474 − 1.22474i −1.22474 + 1.22474i −1.22474 − 1.22474i
0 0 0 −2.22474 0.224745i 0 1.00000i 0 0 0
1729.2 0 0 0 −2.22474 + 0.224745i 0 1.00000i 0 0 0
1729.3 0 0 0 0.224745 2.22474i 0 1.00000i 0 0 0
1729.4 0 0 0 0.224745 + 2.22474i 0 1.00000i 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
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.i 4
3.b odd 2 1 2160.2.f.n 4
4.b odd 2 1 1080.2.f.e 4
5.b even 2 1 inner 2160.2.f.i 4
12.b even 2 1 1080.2.f.f yes 4
15.d odd 2 1 2160.2.f.n 4
20.d odd 2 1 1080.2.f.e 4
20.e even 4 1 5400.2.a.bz 2
20.e even 4 1 5400.2.a.cf 2
60.h even 2 1 1080.2.f.f yes 4
60.l odd 4 1 5400.2.a.bw 2
60.l odd 4 1 5400.2.a.cc 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1080.2.f.e 4 4.b odd 2 1
1080.2.f.e 4 20.d odd 2 1
1080.2.f.f yes 4 12.b even 2 1
1080.2.f.f yes 4 60.h even 2 1
2160.2.f.i 4 1.a even 1 1 trivial
2160.2.f.i 4 5.b even 2 1 inner
2160.2.f.n 4 3.b odd 2 1
2160.2.f.n 4 15.d odd 2 1
5400.2.a.bw 2 60.l odd 4 1
5400.2.a.bz 2 20.e even 4 1
5400.2.a.cc 2 60.l odd 4 1
5400.2.a.cf 2 20.e even 4 1

## Hecke kernels

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

 $$T_{7}^{2} + 1$$ T7^2 + 1 $$T_{11}^{2} + 4T_{11} - 2$$ T11^2 + 4*T11 - 2 $$T_{13}^{4} + 50T_{13}^{2} + 529$$ T13^4 + 50*T13^2 + 529

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4} + 4 T^{3} + \cdots + 25$$
$7$ $$(T^{2} + 1)^{2}$$
$11$ $$(T^{2} + 4 T - 2)^{2}$$
$13$ $$T^{4} + 50T^{2} + 529$$
$17$ $$(T^{2} + 24)^{2}$$
$19$ $$(T^{2} + 2 T - 23)^{2}$$
$23$ $$T^{4} + 20T^{2} + 4$$
$29$ $$(T^{2} + 4 T - 2)^{2}$$
$31$ $$(T + 6)^{4}$$
$37$ $$T^{4} + 98T^{2} + 1$$
$41$ $$(T^{2} - 4 T - 50)^{2}$$
$43$ $$(T^{2} + 36)^{2}$$
$47$ $$(T^{2} + 24)^{2}$$
$53$ $$T^{4} + 20T^{2} + 4$$
$59$ $$(T^{2} - 24)^{2}$$
$61$ $$(T^{2} + 2 T - 95)^{2}$$
$67$ $$T^{4} + 242T^{2} + 5041$$
$71$ $$(T^{2} + 28 T + 190)^{2}$$
$73$ $$T^{4} + 50T^{2} + 529$$
$79$ $$(T^{2} - 2 T - 23)^{2}$$
$83$ $$T^{4} + 212T^{2} + 8836$$
$89$ $$(T + 12)^{4}$$
$97$ $$T^{4} + 290T^{2} + 9409$$