Properties

 Label 2160.2.by.a Level $2160$ Weight $2$ Character orbit 2160.by Analytic conductor $17.248$ 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] = [2160,2,Mod(289,2160)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2160, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 4, 3]))

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

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

chi := DirichletCharacter("2160.289");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2160 = 2^{4} \cdot 3^{3} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2160.by (of order $$6$$, degree $$2$$, 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 360) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - 2 \zeta_{12}^{3} + \cdots + 2 \zeta_{12}) q^{5}+ \cdots + \zeta_{12} q^{7}+O(q^{10})$$ q + (-2*z^3 - z^2 + 2*z) * q^5 + z * q^7 $$q + ( - 2 \zeta_{12}^{3} + \cdots + 2 \zeta_{12}) q^{5}+ \cdots + 16 \zeta_{12} q^{97}+O(q^{100})$$ q + (-2*z^3 - z^2 + 2*z) * q^5 + z * q^7 + (2*z^2 - 2) * q^11 + (2*z^3 - 2*z) * q^13 - 6*z^3 * q^17 + 2 * q^19 + (-z^3 + z) * q^23 + (-3*z^2 - 4*z + 3) * q^25 + (-7*z^2 + 7) * q^29 - 6*z^2 * q^31 + (-z^3 + 2) * q^35 + 2*z^3 * q^37 + 5*z^2 * q^41 + 12*z * q^43 + 9*z * q^47 - 6*z^2 * q^49 - 8*z^3 * q^53 + (4*z^3 + 2) * q^55 - 12*z^2 * q^59 + (-7*z^2 + 7) * q^61 + (4*z^2 + 2*z - 4) * q^65 + (-5*z^3 + 5*z) * q^67 - 10 * q^71 - 4*z^3 * q^73 + (2*z^3 - 2*z) * q^77 + (4*z^2 - 4) * q^79 - 5*z * q^83 + (6*z^3 - 12*z^2 - 6*z) * q^85 - 15 * q^89 - 2 * q^91 + (-4*z^3 - 2*z^2 + 4*z) * q^95 + 16*z * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{5}+O(q^{10})$$ 4 * q - 2 * q^5 $$4 q - 2 q^{5} - 4 q^{11} + 8 q^{19} + 6 q^{25} + 14 q^{29} - 12 q^{31} + 8 q^{35} + 10 q^{41} - 12 q^{49} + 8 q^{55} - 24 q^{59} + 14 q^{61} - 8 q^{65} - 40 q^{71} - 8 q^{79} - 24 q^{85} - 60 q^{89} - 8 q^{91} - 4 q^{95}+O(q^{100})$$ 4 * q - 2 * q^5 - 4 * q^11 + 8 * q^19 + 6 * q^25 + 14 * q^29 - 12 * q^31 + 8 * q^35 + 10 * q^41 - 12 * q^49 + 8 * q^55 - 24 * q^59 + 14 * q^61 - 8 * q^65 - 40 * q^71 - 8 * q^79 - 24 * q^85 - 60 * q^89 - 8 * q^91 - 4 * q^95

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2160\mathbb{Z}\right)^\times$$.

 $$n$$ $$271$$ $$1297$$ $$1621$$ $$2081$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$ $$-\zeta_{12}^{2}$$

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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
289.1
 −0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i
0 0 0 −2.23205 + 0.133975i 0 −0.866025 0.500000i 0 0 0
289.2 0 0 0 1.23205 1.86603i 0 0.866025 + 0.500000i 0 0 0
1009.1 0 0 0 −2.23205 0.133975i 0 −0.866025 + 0.500000i 0 0 0
1009.2 0 0 0 1.23205 + 1.86603i 0 0.866025 0.500000i 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
9.c even 3 1 inner
45.j even 6 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2160.2.by.a 4
3.b odd 2 1 720.2.by.b 4
4.b odd 2 1 1080.2.bi.a 4
5.b even 2 1 inner 2160.2.by.a 4
9.c even 3 1 inner 2160.2.by.a 4
9.d odd 6 1 720.2.by.b 4
12.b even 2 1 360.2.bi.a 4
15.d odd 2 1 720.2.by.b 4
20.d odd 2 1 1080.2.bi.a 4
36.f odd 6 1 1080.2.bi.a 4
36.f odd 6 1 3240.2.f.e 2
36.h even 6 1 360.2.bi.a 4
36.h even 6 1 3240.2.f.b 2
45.h odd 6 1 720.2.by.b 4
45.j even 6 1 inner 2160.2.by.a 4
60.h even 2 1 360.2.bi.a 4
180.n even 6 1 360.2.bi.a 4
180.n even 6 1 3240.2.f.b 2
180.p odd 6 1 1080.2.bi.a 4
180.p odd 6 1 3240.2.f.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.2.bi.a 4 12.b even 2 1
360.2.bi.a 4 36.h even 6 1
360.2.bi.a 4 60.h even 2 1
360.2.bi.a 4 180.n even 6 1
720.2.by.b 4 3.b odd 2 1
720.2.by.b 4 9.d odd 6 1
720.2.by.b 4 15.d odd 2 1
720.2.by.b 4 45.h odd 6 1
1080.2.bi.a 4 4.b odd 2 1
1080.2.bi.a 4 20.d odd 2 1
1080.2.bi.a 4 36.f odd 6 1
1080.2.bi.a 4 180.p odd 6 1
2160.2.by.a 4 1.a even 1 1 trivial
2160.2.by.a 4 5.b even 2 1 inner
2160.2.by.a 4 9.c even 3 1 inner
2160.2.by.a 4 45.j even 6 1 inner
3240.2.f.b 2 36.h even 6 1
3240.2.f.b 2 180.n even 6 1
3240.2.f.e 2 36.f odd 6 1
3240.2.f.e 2 180.p odd 6 1

Hecke kernels

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

 $$T_{7}^{4} - T_{7}^{2} + 1$$ T7^4 - T7^2 + 1 $$T_{11}^{2} + 2T_{11} + 4$$ T11^2 + 2*T11 + 4

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4} + 2 T^{3} + \cdots + 25$$
$7$ $$T^{4} - T^{2} + 1$$
$11$ $$(T^{2} + 2 T + 4)^{2}$$
$13$ $$T^{4} - 4T^{2} + 16$$
$17$ $$(T^{2} + 36)^{2}$$
$19$ $$(T - 2)^{4}$$
$23$ $$T^{4} - T^{2} + 1$$
$29$ $$(T^{2} - 7 T + 49)^{2}$$
$31$ $$(T^{2} + 6 T + 36)^{2}$$
$37$ $$(T^{2} + 4)^{2}$$
$41$ $$(T^{2} - 5 T + 25)^{2}$$
$43$ $$T^{4} - 144 T^{2} + 20736$$
$47$ $$T^{4} - 81T^{2} + 6561$$
$53$ $$(T^{2} + 64)^{2}$$
$59$ $$(T^{2} + 12 T + 144)^{2}$$
$61$ $$(T^{2} - 7 T + 49)^{2}$$
$67$ $$T^{4} - 25T^{2} + 625$$
$71$ $$(T + 10)^{4}$$
$73$ $$(T^{2} + 16)^{2}$$
$79$ $$(T^{2} + 4 T + 16)^{2}$$
$83$ $$T^{4} - 25T^{2} + 625$$
$89$ $$(T + 15)^{4}$$
$97$ $$T^{4} - 256 T^{2} + 65536$$