# Properties

 Label 2160.3.c.j Level $2160$ Weight $3$ Character orbit 2160.c Analytic conductor $58.856$ 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] = [2160,3,Mod(1889,2160)]

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, 1, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("2160.1889");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2160 = 2^{4} \cdot 3^{3} \cdot 5$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 2160.c (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: $$58.8557371018$$ 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: $$\Z[a_1, \ldots, a_{17}]$$ 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 primitive root of unity $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - 3 \zeta_{8}^{3} + 4 \zeta_{8}) q^{5} - 11 \zeta_{8}^{2} q^{7} +O(q^{10})$$ q + (-3*z^3 + 4*z) * q^5 - 11*z^2 * q^7 $$q + ( - 3 \zeta_{8}^{3} + 4 \zeta_{8}) q^{5} - 11 \zeta_{8}^{2} q^{7} + (5 \zeta_{8}^{3} + 5 \zeta_{8}) q^{11} + 15 \zeta_{8}^{2} q^{13} + (16 \zeta_{8}^{3} - 16 \zeta_{8}) q^{17} + 3 q^{19} + (13 \zeta_{8}^{3} - 13 \zeta_{8}) q^{23} + (7 \zeta_{8}^{2} + 24) q^{25} + (9 \zeta_{8}^{3} + 9 \zeta_{8}) q^{29} + 8 q^{31} + ( - 44 \zeta_{8}^{3} - 33 \zeta_{8}) q^{35} + 65 \zeta_{8}^{2} q^{37} + (55 \zeta_{8}^{3} + 55 \zeta_{8}) q^{41} + 32 \zeta_{8}^{2} q^{43} + (40 \zeta_{8}^{3} - 40 \zeta_{8}) q^{47} - 72 q^{49} + (9 \zeta_{8}^{3} - 9 \zeta_{8}) q^{53} + (35 \zeta_{8}^{2} - 5) q^{55} + (56 \zeta_{8}^{3} + 56 \zeta_{8}) q^{59} + 95 q^{61} + (60 \zeta_{8}^{3} + 45 \zeta_{8}) q^{65} - 19 \zeta_{8}^{2} q^{67} + ( - 3 \zeta_{8}^{3} - 3 \zeta_{8}) q^{71} - 119 \zeta_{8}^{2} q^{73} + ( - 55 \zeta_{8}^{3} + 55 \zeta_{8}) q^{77} - 99 q^{79} + ( - 77 \zeta_{8}^{3} + 77 \zeta_{8}) q^{83} + ( - 16 \zeta_{8}^{2} - 112) q^{85} + ( - 64 \zeta_{8}^{3} - 64 \zeta_{8}) q^{89} + 165 q^{91} + ( - 9 \zeta_{8}^{3} + 12 \zeta_{8}) q^{95} - 95 \zeta_{8}^{2} q^{97} +O(q^{100})$$ q + (-3*z^3 + 4*z) * q^5 - 11*z^2 * q^7 + (5*z^3 + 5*z) * q^11 + 15*z^2 * q^13 + (16*z^3 - 16*z) * q^17 + 3 * q^19 + (13*z^3 - 13*z) * q^23 + (7*z^2 + 24) * q^25 + (9*z^3 + 9*z) * q^29 + 8 * q^31 + (-44*z^3 - 33*z) * q^35 + 65*z^2 * q^37 + (55*z^3 + 55*z) * q^41 + 32*z^2 * q^43 + (40*z^3 - 40*z) * q^47 - 72 * q^49 + (9*z^3 - 9*z) * q^53 + (35*z^2 - 5) * q^55 + (56*z^3 + 56*z) * q^59 + 95 * q^61 + (60*z^3 + 45*z) * q^65 - 19*z^2 * q^67 + (-3*z^3 - 3*z) * q^71 - 119*z^2 * q^73 + (-55*z^3 + 55*z) * q^77 - 99 * q^79 + (-77*z^3 + 77*z) * q^83 + (-16*z^2 - 112) * q^85 + (-64*z^3 - 64*z) * q^89 + 165 * q^91 + (-9*z^3 + 12*z) * q^95 - 95*z^2 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q + 12 q^{19} + 96 q^{25} + 32 q^{31} - 288 q^{49} - 20 q^{55} + 380 q^{61} - 396 q^{79} - 448 q^{85} + 660 q^{91}+O(q^{100})$$ 4 * q + 12 * q^19 + 96 * q^25 + 32 * q^31 - 288 * q^49 - 20 * q^55 + 380 * q^61 - 396 * q^79 - 448 * q^85 + 660 * q^91

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1889.1
 −0.707107 − 0.707107i −0.707107 + 0.707107i 0.707107 − 0.707107i 0.707107 + 0.707107i
0 0 0 −4.94975 0.707107i 0 11.0000i 0 0 0
1889.2 0 0 0 −4.94975 + 0.707107i 0 11.0000i 0 0 0
1889.3 0 0 0 4.94975 0.707107i 0 11.0000i 0 0 0
1889.4 0 0 0 4.94975 + 0.707107i 0 11.0000i 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 2160.3.c.j 4
3.b odd 2 1 inner 2160.3.c.j 4
4.b odd 2 1 270.3.b.b 4
5.b even 2 1 inner 2160.3.c.j 4
12.b even 2 1 270.3.b.b 4
15.d odd 2 1 inner 2160.3.c.j 4
20.d odd 2 1 270.3.b.b 4
20.e even 4 1 1350.3.d.b 2
20.e even 4 1 1350.3.d.j 2
36.f odd 6 2 810.3.j.b 8
36.h even 6 2 810.3.j.b 8
60.h even 2 1 270.3.b.b 4
60.l odd 4 1 1350.3.d.b 2
60.l odd 4 1 1350.3.d.j 2
180.n even 6 2 810.3.j.b 8
180.p odd 6 2 810.3.j.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
270.3.b.b 4 4.b odd 2 1
270.3.b.b 4 12.b even 2 1
270.3.b.b 4 20.d odd 2 1
270.3.b.b 4 60.h even 2 1
810.3.j.b 8 36.f odd 6 2
810.3.j.b 8 36.h even 6 2
810.3.j.b 8 180.n even 6 2
810.3.j.b 8 180.p odd 6 2
1350.3.d.b 2 20.e even 4 1
1350.3.d.b 2 60.l odd 4 1
1350.3.d.j 2 20.e even 4 1
1350.3.d.j 2 60.l odd 4 1
2160.3.c.j 4 1.a even 1 1 trivial
2160.3.c.j 4 3.b odd 2 1 inner
2160.3.c.j 4 5.b even 2 1 inner
2160.3.c.j 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 $$S_{3}^{\mathrm{new}}(2160, [\chi])$$:

 $$T_{7}^{2} + 121$$ T7^2 + 121 $$T_{17}^{2} - 512$$ T17^2 - 512

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4} - 48T^{2} + 625$$
$7$ $$(T^{2} + 121)^{2}$$
$11$ $$(T^{2} + 50)^{2}$$
$13$ $$(T^{2} + 225)^{2}$$
$17$ $$(T^{2} - 512)^{2}$$
$19$ $$(T - 3)^{4}$$
$23$ $$(T^{2} - 338)^{2}$$
$29$ $$(T^{2} + 162)^{2}$$
$31$ $$(T - 8)^{4}$$
$37$ $$(T^{2} + 4225)^{2}$$
$41$ $$(T^{2} + 6050)^{2}$$
$43$ $$(T^{2} + 1024)^{2}$$
$47$ $$(T^{2} - 3200)^{2}$$
$53$ $$(T^{2} - 162)^{2}$$
$59$ $$(T^{2} + 6272)^{2}$$
$61$ $$(T - 95)^{4}$$
$67$ $$(T^{2} + 361)^{2}$$
$71$ $$(T^{2} + 18)^{2}$$
$73$ $$(T^{2} + 14161)^{2}$$
$79$ $$(T + 99)^{4}$$
$83$ $$(T^{2} - 11858)^{2}$$
$89$ $$(T^{2} + 8192)^{2}$$
$97$ $$(T^{2} + 9025)^{2}$$