# Properties

 Label 2160.2.o.a Level $2160$ Weight $2$ Character orbit 2160.o 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(2159,2160)]

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([1, 0, 1, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("2160.2159");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2160 = 2^{4} \cdot 3^{3} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2160.o (of order $$2$$, degree $$1$$, 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: $$\Q(\sqrt{-3}, \sqrt{-11})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 2x^{2} - 3x + 9$$ x^4 - x^3 - 2*x^2 - 3*x + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{3} + \beta_{2} - 1) q^{5} + ( - \beta_{3} - \beta_1 - 2) q^{7}+O(q^{10})$$ q + (-b3 + b2 - 1) * q^5 + (-b3 - b1 - 2) * q^7 $$q + ( - \beta_{3} + \beta_{2} - 1) q^{5} + ( - \beta_{3} - \beta_1 - 2) q^{7} + ( - \beta_{3} - \beta_1 - 2) q^{11} + 2 \beta_{2} q^{13} + ( - \beta_{3} - \beta_1 + 1) q^{17} + ( - 3 \beta_{3} + \beta_{2} + 3 \beta_1) q^{19} + (3 \beta_{3} - \beta_{2} - 3 \beta_1) q^{23} + (\beta_{2} + 3 \beta_1 + 1) q^{25} + ( - 2 \beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{29} - \beta_{2} q^{31} + (\beta_{2} + 3 \beta_1 + 6) q^{35} + 4 \beta_{2} q^{37} + (2 \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{41} + 6 q^{43} - 2 \beta_{2} q^{47} + (3 \beta_{3} + 3 \beta_1 + 5) q^{49} - 3 q^{53} + (\beta_{2} + 3 \beta_1 + 6) q^{55} + (2 \beta_{3} + 2 \beta_1 - 2) q^{59} + (3 \beta_{3} + 3 \beta_1 + 7) q^{61} + ( - 2 \beta_{3} + 4 \beta_1 - 4) q^{65} + ( - 2 \beta_{3} - 2 \beta_1 + 8) q^{67} + ( - 4 \beta_{3} - 4 \beta_1 - 2) q^{71} + ( - 9 \beta_{3} + 4 \beta_{2} + 9 \beta_1) q^{73} + (3 \beta_{3} + 3 \beta_1 + 12) q^{77} + (3 \beta_{3} - 3 \beta_{2} - 3 \beta_1) q^{79} + ( - 6 \beta_{3} + 5 \beta_{2} + 6 \beta_1) q^{83} + ( - 3 \beta_{3} + 4 \beta_{2} + \cdots + 3) q^{85}+ \cdots + ( - 3 \beta_{3} + 6 \beta_{2} + 3 \beta_1) q^{97}+O(q^{100})$$ q + (-b3 + b2 - 1) * q^5 + (-b3 - b1 - 2) * q^7 + (-b3 - b1 - 2) * q^11 + 2*b2 * q^13 + (-b3 - b1 + 1) * q^17 + (-3*b3 + b2 + 3*b1) * q^19 + (3*b3 - b2 - 3*b1) * q^23 + (b2 + 3*b1 + 1) * q^25 + (-2*b3 - 2*b2 + 2*b1) * q^29 - b2 * q^31 + (b2 + 3*b1 + 6) * q^35 + 4*b2 * q^37 + (2*b3 + 2*b2 - 2*b1) * q^41 + 6 * q^43 - 2*b2 * q^47 + (3*b3 + 3*b1 + 5) * q^49 - 3 * q^53 + (b2 + 3*b1 + 6) * q^55 + (2*b3 + 2*b1 - 2) * q^59 + (3*b3 + 3*b1 + 7) * q^61 + (-2*b3 + 4*b1 - 4) * q^65 + (-2*b3 - 2*b1 + 8) * q^67 + (-4*b3 - 4*b1 - 2) * q^71 + (-9*b3 + 4*b2 + 9*b1) * q^73 + (3*b3 + 3*b1 + 12) * q^77 + (3*b3 - 3*b2 - 3*b1) * q^79 + (-6*b3 + 5*b2 + 6*b1) * q^83 + (-3*b3 + 4*b2 + 3*b1 + 3) * q^85 + (4*b3 - 2*b2 - 4*b1) * q^89 + (-6*b3 + 6*b1) * q^91 + (5*b3 + 3*b2 - b1 - 2) * q^95 + (-3*b3 + 6*b2 + 3*b1) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 3 q^{5} - 6 q^{7}+O(q^{10})$$ 4 * q - 3 * q^5 - 6 * q^7 $$4 q - 3 q^{5} - 6 q^{7} - 6 q^{11} + 6 q^{17} + q^{25} + 21 q^{35} + 24 q^{43} + 14 q^{49} - 12 q^{53} + 21 q^{55} - 12 q^{59} + 22 q^{61} - 18 q^{65} + 36 q^{67} + 42 q^{77} + 12 q^{85} - 12 q^{95}+O(q^{100})$$ 4 * q - 3 * q^5 - 6 * q^7 - 6 * q^11 + 6 * q^17 + q^25 + 21 * q^35 + 24 * q^43 + 14 * q^49 - 12 * q^53 + 21 * q^55 - 12 * q^59 + 22 * q^61 - 18 * q^65 + 36 * q^67 + 42 * q^77 + 12 * q^85 - 12 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 2x^{2} - 3x + 9$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} + 2\nu^{2} + 4\nu - 9 ) / 6$$ (v^3 + 2*v^2 + 4*v - 9) / 6 $$\beta_{2}$$ $$=$$ $$( -\nu^{3} - 2\nu^{2} + 2\nu + 6 ) / 3$$ (-v^3 - 2*v^2 + 2*v + 6) / 3 $$\beta_{3}$$ $$=$$ $$( -\nu^{3} + 2\nu + 3 ) / 2$$ (-v^3 + 2*v + 3) / 2
 $$\nu$$ $$=$$ $$( \beta_{2} + 2\beta _1 + 1 ) / 2$$ (b2 + 2*b1 + 1) / 2 $$\nu^{2}$$ $$=$$ $$( 2\beta_{3} - 3\beta_{2} + 3 ) / 2$$ (2*b3 - 3*b2 + 3) / 2 $$\nu^{3}$$ $$=$$ $$-2\beta_{3} + \beta_{2} + 2\beta _1 + 4$$ -2*b3 + b2 + 2*b1 + 4

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2159.1
 1.68614 + 0.396143i 1.68614 − 0.396143i −1.18614 − 1.26217i −1.18614 + 1.26217i
0 0 0 −2.18614 0.469882i 0 −4.37228 0 0 0
2159.2 0 0 0 −2.18614 + 0.469882i 0 −4.37228 0 0 0
2159.3 0 0 0 0.686141 2.12819i 0 1.37228 0 0 0
2159.4 0 0 0 0.686141 + 2.12819i 0 1.37228 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
60.h 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.o.a 4
3.b odd 2 1 2160.2.o.c yes 4
4.b odd 2 1 2160.2.o.b yes 4
5.b even 2 1 2160.2.o.d yes 4
12.b even 2 1 2160.2.o.d yes 4
15.d odd 2 1 2160.2.o.b yes 4
20.d odd 2 1 2160.2.o.c yes 4
60.h even 2 1 inner 2160.2.o.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2160.2.o.a 4 1.a even 1 1 trivial
2160.2.o.a 4 60.h even 2 1 inner
2160.2.o.b yes 4 4.b odd 2 1
2160.2.o.b yes 4 15.d odd 2 1
2160.2.o.c yes 4 3.b odd 2 1
2160.2.o.c yes 4 20.d odd 2 1
2160.2.o.d yes 4 5.b even 2 1
2160.2.o.d yes 4 12.b even 2 1

## Hecke kernels

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

 $$T_{7}^{2} + 3T_{7} - 6$$ T7^2 + 3*T7 - 6 $$T_{11}^{2} + 3T_{11} - 6$$ T11^2 + 3*T11 - 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4} + 3 T^{3} + \cdots + 25$$
$7$ $$(T^{2} + 3 T - 6)^{2}$$
$11$ $$(T^{2} + 3 T - 6)^{2}$$
$13$ $$(T^{2} + 12)^{2}$$
$17$ $$(T^{2} - 3 T - 6)^{2}$$
$19$ $$T^{4} + 51T^{2} + 576$$
$23$ $$T^{4} + 51T^{2} + 576$$
$29$ $$T^{4} + 76T^{2} + 256$$
$31$ $$(T^{2} + 3)^{2}$$
$37$ $$(T^{2} + 48)^{2}$$
$41$ $$T^{4} + 76T^{2} + 256$$
$43$ $$(T - 6)^{4}$$
$47$ $$(T^{2} + 12)^{2}$$
$53$ $$(T + 3)^{4}$$
$59$ $$(T^{2} + 6 T - 24)^{2}$$
$61$ $$(T^{2} - 11 T - 44)^{2}$$
$67$ $$(T^{2} - 18 T + 48)^{2}$$
$71$ $$(T^{2} - 132)^{2}$$
$73$ $$T^{4} + 447 T^{2} + 49284$$
$79$ $$T^{4} + 63T^{2} + 324$$
$83$ $$T^{4} + 222T^{2} + 7569$$
$89$ $$(T^{2} + 44)^{2}$$
$97$ $$T^{4} + 171T^{2} + 1296$$