# Properties

 Label 2160.3.l.e Level $2160$ Weight $3$ Character orbit 2160.l Analytic conductor $58.856$ 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] = [2160,3,Mod(161,2160)]

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, 0]))

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

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

chi := DirichletCharacter("2160.161");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2160 = 2^{4} \cdot 3^{3} \cdot 5$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 2160.l (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: $$\Q(i, \sqrt{5})$$ 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: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}\cdot 3^{2}$$ Twist minimal: no (minimal twist has level 540) 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{5} + (\beta_{3} - 5) q^{7}+O(q^{10})$$ q + b1 * q^5 + (b3 - 5) * q^7 $$q + \beta_1 q^{5} + (\beta_{3} - 5) q^{7} + (\beta_{2} + 3 \beta_1) q^{11} + ( - \beta_{3} - 7) q^{13} - 5 \beta_{2} q^{17} + (2 \beta_{3} + 1) q^{19} + ( - 7 \beta_{2} - 6 \beta_1) q^{23} - 5 q^{25} + (\beta_{2} - 3 \beta_1) q^{29} + (2 \beta_{3} + 7) q^{31} + ( - 5 \beta_{2} - 5 \beta_1) q^{35} + ( - 6 \beta_{3} - 16) q^{37} + (3 \beta_{2} - 15 \beta_1) q^{41} + (7 \beta_{3} - 29) q^{43} + (4 \beta_{2} + 6 \beta_1) q^{47} + ( - 10 \beta_{3} + 21) q^{49} + ( - \beta_{2} + 30 \beta_1) q^{53} + (\beta_{3} - 15) q^{55} + (17 \beta_{2} + 9 \beta_1) q^{59} + ( - 12 \beta_{3} - 1) q^{61} + (5 \beta_{2} - 7 \beta_1) q^{65} - 14 q^{67} + ( - 15 \beta_{2} - 21 \beta_1) q^{71} + ( - 3 \beta_{3} - 7) q^{73} + ( - 20 \beta_{2} - 24 \beta_1) q^{77} + ( - 12 \beta_{3} + 67) q^{79} + (11 \beta_{2} + 48 \beta_1) q^{83} - 5 \beta_{3} q^{85} + ( - 3 \beta_{2} + 51 \beta_1) q^{89} + ( - 2 \beta_{3} - 10) q^{91} + ( - 10 \beta_{2} + \beta_1) q^{95} + ( - 2 \beta_{3} + 2) q^{97}+O(q^{100})$$ q + b1 * q^5 + (b3 - 5) * q^7 + (b2 + 3*b1) * q^11 + (-b3 - 7) * q^13 - 5*b2 * q^17 + (2*b3 + 1) * q^19 + (-7*b2 - 6*b1) * q^23 - 5 * q^25 + (b2 - 3*b1) * q^29 + (2*b3 + 7) * q^31 + (-5*b2 - 5*b1) * q^35 + (-6*b3 - 16) * q^37 + (3*b2 - 15*b1) * q^41 + (7*b3 - 29) * q^43 + (4*b2 + 6*b1) * q^47 + (-10*b3 + 21) * q^49 + (-b2 + 30*b1) * q^53 + (b3 - 15) * q^55 + (17*b2 + 9*b1) * q^59 + (-12*b3 - 1) * q^61 + (5*b2 - 7*b1) * q^65 - 14 * q^67 + (-15*b2 - 21*b1) * q^71 + (-3*b3 - 7) * q^73 + (-20*b2 - 24*b1) * q^77 + (-12*b3 + 67) * q^79 + (11*b2 + 48*b1) * q^83 - 5*b3 * q^85 + (-3*b2 + 51*b1) * q^89 + (-2*b3 - 10) * q^91 + (-10*b2 + b1) * q^95 + (-2*b3 + 2) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 20 q^{7}+O(q^{10})$$ 4 * q - 20 * q^7 $$4 q - 20 q^{7} - 28 q^{13} + 4 q^{19} - 20 q^{25} + 28 q^{31} - 64 q^{37} - 116 q^{43} + 84 q^{49} - 60 q^{55} - 4 q^{61} - 56 q^{67} - 28 q^{73} + 268 q^{79} - 40 q^{91} + 8 q^{97}+O(q^{100})$$ 4 * q - 20 * q^7 - 28 * q^13 + 4 * q^19 - 20 * q^25 + 28 * q^31 - 64 * q^37 - 116 * q^43 + 84 * q^49 - 60 * q^55 - 4 * q^61 - 56 * q^67 - 28 * q^73 + 268 * q^79 - 40 * q^91 + 8 * q^97

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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
161.1
 − 1.61803i − 0.618034i 1.61803i 0.618034i
0 0 0 2.23607i 0 −11.7082 0 0 0
161.2 0 0 0 2.23607i 0 1.70820 0 0 0
161.3 0 0 0 2.23607i 0 −11.7082 0 0 0
161.4 0 0 0 2.23607i 0 1.70820 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2160.3.l.e 4
3.b odd 2 1 inner 2160.3.l.e 4
4.b odd 2 1 540.3.g.d 4
12.b even 2 1 540.3.g.d 4
20.d odd 2 1 2700.3.g.n 4
20.e even 4 1 2700.3.b.g 4
20.e even 4 1 2700.3.b.l 4
36.f odd 6 2 1620.3.o.e 8
36.h even 6 2 1620.3.o.e 8
60.h even 2 1 2700.3.g.n 4
60.l odd 4 1 2700.3.b.g 4
60.l odd 4 1 2700.3.b.l 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
540.3.g.d 4 4.b odd 2 1
540.3.g.d 4 12.b even 2 1
1620.3.o.e 8 36.f odd 6 2
1620.3.o.e 8 36.h even 6 2
2160.3.l.e 4 1.a even 1 1 trivial
2160.3.l.e 4 3.b odd 2 1 inner
2700.3.b.g 4 20.e even 4 1
2700.3.b.g 4 60.l odd 4 1
2700.3.b.l 4 20.e even 4 1
2700.3.b.l 4 60.l odd 4 1
2700.3.g.n 4 20.d odd 2 1
2700.3.g.n 4 60.h even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{2} + 10T_{7} - 20$$ acting on $$S_{3}^{\mathrm{new}}(2160, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$(T^{2} + 5)^{2}$$
$7$ $$(T^{2} + 10 T - 20)^{2}$$
$11$ $$T^{4} + 108T^{2} + 1296$$
$13$ $$(T^{2} + 14 T + 4)^{2}$$
$17$ $$(T^{2} + 225)^{2}$$
$19$ $$(T^{2} - 2 T - 179)^{2}$$
$23$ $$T^{4} + 1242 T^{2} + 68121$$
$29$ $$T^{4} + 108T^{2} + 1296$$
$31$ $$(T^{2} - 14 T - 131)^{2}$$
$37$ $$(T^{2} + 32 T - 1364)^{2}$$
$41$ $$T^{4} + 2412 T^{2} + 1089936$$
$43$ $$(T^{2} + 58 T - 1364)^{2}$$
$47$ $$T^{4} + 648T^{2} + 1296$$
$53$ $$T^{4} + 9018 T^{2} + 20169081$$
$59$ $$T^{4} + 6012 T^{2} + 4822416$$
$61$ $$(T^{2} + 2 T - 6479)^{2}$$
$67$ $$(T + 14)^{4}$$
$71$ $$T^{4} + 8460 T^{2} + 32400$$
$73$ $$(T^{2} + 14 T - 356)^{2}$$
$79$ $$(T^{2} - 134 T - 1991)^{2}$$
$83$ $$T^{4} + 25218 T^{2} + 108805761$$
$89$ $$T^{4} + 26172 T^{2} + 167029776$$
$97$ $$(T^{2} - 4 T - 176)^{2}$$