# Properties

 Label 2160.3.c.k 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: $$\Q(i, \sqrt{10})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 25$$ x^4 + 25 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.

 $$f(q)$$ $$=$$ $$q + (\beta_{3} - \beta_{2}) q^{5} + \beta_1 q^{7}+O(q^{10})$$ q + (b3 - b2) * q^5 + b1 * q^7 $$q + (\beta_{3} - \beta_{2}) q^{5} + \beta_1 q^{7} + (\beta_{3} - 2 \beta_{2}) q^{11} + 7 \beta_1 q^{13} + 4 \beta_{3} q^{17} + 31 q^{19} - 7 \beta_{3} q^{23} + ( - 5 \beta_1 - 20) q^{25} + (5 \beta_{3} - 10 \beta_{2}) q^{29} + 16 q^{31} + (4 \beta_{3} + \beta_{2}) q^{35} + 9 \beta_1 q^{37} + ( - 5 \beta_{3} + 10 \beta_{2}) q^{41} + 16 \beta_1 q^{43} - 4 \beta_{3} q^{47} + 40 q^{49} + 13 \beta_{3} q^{53} + ( - 5 \beta_1 - 45) q^{55} + ( - 4 \beta_{3} + 8 \beta_{2}) q^{59} - q^{61} + (28 \beta_{3} + 7 \beta_{2}) q^{65} - 7 \beta_1 q^{67} + ( - 3 \beta_{3} + 6 \beta_{2}) q^{71} + 9 \beta_1 q^{73} + 9 \beta_{3} q^{77} + q^{79} + 35 \beta_{3} q^{83} + ( - 20 \beta_1 + 20) q^{85} + (12 \beta_{3} - 24 \beta_{2}) q^{89} - 63 q^{91} + (31 \beta_{3} - 31 \beta_{2}) q^{95} - 31 \beta_1 q^{97}+O(q^{100})$$ q + (b3 - b2) * q^5 + b1 * q^7 + (b3 - 2*b2) * q^11 + 7*b1 * q^13 + 4*b3 * q^17 + 31 * q^19 - 7*b3 * q^23 + (-5*b1 - 20) * q^25 + (5*b3 - 10*b2) * q^29 + 16 * q^31 + (4*b3 + b2) * q^35 + 9*b1 * q^37 + (-5*b3 + 10*b2) * q^41 + 16*b1 * q^43 - 4*b3 * q^47 + 40 * q^49 + 13*b3 * q^53 + (-5*b1 - 45) * q^55 + (-4*b3 + 8*b2) * q^59 - q^61 + (28*b3 + 7*b2) * q^65 - 7*b1 * q^67 + (-3*b3 + 6*b2) * q^71 + 9*b1 * q^73 + 9*b3 * q^77 + q^79 + 35*b3 * q^83 + (-20*b1 + 20) * q^85 + (12*b3 - 24*b2) * q^89 - 63 * q^91 + (31*b3 - 31*b2) * q^95 - 31*b1 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q + 124 q^{19} - 80 q^{25} + 64 q^{31} + 160 q^{49} - 180 q^{55} - 4 q^{61} + 4 q^{79} + 80 q^{85} - 252 q^{91}+O(q^{100})$$ 4 * q + 124 * q^19 - 80 * q^25 + 64 * q^31 + 160 * q^49 - 180 * q^55 - 4 * q^61 + 4 * q^79 + 80 * q^85 - 252 * q^91

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

 $$\beta_{1}$$ $$=$$ $$( 3\nu^{2} ) / 5$$ (3*v^2) / 5 $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 10\nu ) / 5$$ (v^3 + 10*v) / 5 $$\beta_{3}$$ $$=$$ $$( -\nu^{3} + 5\nu ) / 5$$ (-v^3 + 5*v) / 5
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_{2} ) / 3$$ (b3 + b2) / 3 $$\nu^{2}$$ $$=$$ $$( 5\beta_1 ) / 3$$ (5*b1) / 3 $$\nu^{3}$$ $$=$$ $$( -10\beta_{3} + 5\beta_{2} ) / 3$$ (-10*b3 + 5*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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1889.1
 −1.58114 + 1.58114i −1.58114 − 1.58114i 1.58114 + 1.58114i 1.58114 − 1.58114i
0 0 0 −1.58114 4.74342i 0 3.00000i 0 0 0
1889.2 0 0 0 −1.58114 + 4.74342i 0 3.00000i 0 0 0
1889.3 0 0 0 1.58114 4.74342i 0 3.00000i 0 0 0
1889.4 0 0 0 1.58114 + 4.74342i 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 2160.3.c.k 4
3.b odd 2 1 inner 2160.3.c.k 4
4.b odd 2 1 135.3.d.g 4
5.b even 2 1 inner 2160.3.c.k 4
12.b even 2 1 135.3.d.g 4
15.d odd 2 1 inner 2160.3.c.k 4
20.d odd 2 1 135.3.d.g 4
20.e even 4 1 675.3.c.f 2
20.e even 4 1 675.3.c.g 2
36.f odd 6 2 405.3.h.i 8
36.h even 6 2 405.3.h.i 8
60.h even 2 1 135.3.d.g 4
60.l odd 4 1 675.3.c.f 2
60.l odd 4 1 675.3.c.g 2
180.n even 6 2 405.3.h.i 8
180.p odd 6 2 405.3.h.i 8

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

 $$T_{7}^{2} + 9$$ T7^2 + 9 $$T_{17}^{2} - 160$$ T17^2 - 160

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4} + 40T^{2} + 625$$
$7$ $$(T^{2} + 9)^{2}$$
$11$ $$(T^{2} + 90)^{2}$$
$13$ $$(T^{2} + 441)^{2}$$
$17$ $$(T^{2} - 160)^{2}$$
$19$ $$(T - 31)^{4}$$
$23$ $$(T^{2} - 490)^{2}$$
$29$ $$(T^{2} + 2250)^{2}$$
$31$ $$(T - 16)^{4}$$
$37$ $$(T^{2} + 729)^{2}$$
$41$ $$(T^{2} + 2250)^{2}$$
$43$ $$(T^{2} + 2304)^{2}$$
$47$ $$(T^{2} - 160)^{2}$$
$53$ $$(T^{2} - 1690)^{2}$$
$59$ $$(T^{2} + 1440)^{2}$$
$61$ $$(T + 1)^{4}$$
$67$ $$(T^{2} + 441)^{2}$$
$71$ $$(T^{2} + 810)^{2}$$
$73$ $$(T^{2} + 729)^{2}$$
$79$ $$(T - 1)^{4}$$
$83$ $$(T^{2} - 12250)^{2}$$
$89$ $$(T^{2} + 12960)^{2}$$
$97$ $$(T^{2} + 8649)^{2}$$