Properties

 Label 2160.1.bu.a Level $2160$ Weight $1$ Character orbit 2160.bu Analytic conductor $1.078$ Analytic rank $0$ Dimension $4$ Projective image $D_{6}$ CM discriminant -20 Inner twists $8$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2160,1,Mod(559,2160)]

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([3, 0, 4, 3]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("2160.559");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2160 = 2^{4} \cdot 3^{3} \cdot 5$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 2160.bu (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: $$1.07798042729$$ 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: $$3$$ Twist minimal: no (minimal twist has level 720) Projective image: $$D_{6}$$ Projective field: Galois closure of 6.0.10497600.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q + \zeta_{12}^{4} q^{5} + (\zeta_{12}^{3} + \zeta_{12}) q^{7}+O(q^{10})$$ q + z^4 * q^5 + (z^3 + z) * q^7 $$q + \zeta_{12}^{4} q^{5} + (\zeta_{12}^{3} + \zeta_{12}) q^{7} + (\zeta_{12}^{5} + \zeta_{12}^{3}) q^{23} - \zeta_{12}^{2} q^{25} - \zeta_{12}^{2} q^{29} + (\zeta_{12}^{5} - \zeta_{12}) q^{35} - \zeta_{12}^{4} q^{41} + (\zeta_{12}^{3} + \zeta_{12}) q^{47} + (\zeta_{12}^{4} + \zeta_{12}^{2} - 1) q^{49} - \zeta_{12}^{2} q^{61} + (\zeta_{12}^{5} + \zeta_{12}^{3}) q^{67} + ( - \zeta_{12}^{3} - \zeta_{12}) q^{83} + q^{89} +O(q^{100})$$ q + z^4 * q^5 + (z^3 + z) * q^7 + (z^5 + z^3) * q^23 - z^2 * q^25 - z^2 * q^29 + (z^5 - z) * q^35 - z^4 * q^41 + (z^3 + z) * q^47 + (z^4 + z^2 - 1) * q^49 - z^2 * q^61 + (z^5 + z^3) * q^67 + (-z^3 - z) * q^83 + q^89 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{5}+O(q^{10})$$ 4 * q - 2 * q^5 $$4 q - 2 q^{5} - 2 q^{25} - 2 q^{29} + 2 q^{41} - 4 q^{49} - 2 q^{61} + 4 q^{89}+O(q^{100})$$ 4 * q - 2 * q^5 - 2 * q^25 - 2 * q^29 + 2 * q^41 - 4 * q^49 - 2 * q^61 + 4 * q^89

Character values

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

 $$n$$ $$271$$ $$1297$$ $$1621$$ $$2081$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$ $$\zeta_{12}^{4}$$

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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
559.1
 −0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 − 0.500000i 0.866025 + 0.500000i
0 0 0 −0.500000 0.866025i 0 −0.866025 + 1.50000i 0 0 0
559.2 0 0 0 −0.500000 0.866025i 0 0.866025 1.50000i 0 0 0
1279.1 0 0 0 −0.500000 + 0.866025i 0 −0.866025 1.50000i 0 0 0
1279.2 0 0 0 −0.500000 + 0.866025i 0 0.866025 + 1.50000i 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
20.d odd 2 1 CM by $$\Q(\sqrt{-5})$$
4.b odd 2 1 inner
5.b even 2 1 inner
9.c even 3 1 inner
36.f odd 6 1 inner
45.j even 6 1 inner
180.p odd 6 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2160.1.bu.a 4
3.b odd 2 1 720.1.bu.a 4
4.b odd 2 1 inner 2160.1.bu.a 4
5.b even 2 1 inner 2160.1.bu.a 4
9.c even 3 1 inner 2160.1.bu.a 4
9.d odd 6 1 720.1.bu.a 4
12.b even 2 1 720.1.bu.a 4
15.d odd 2 1 720.1.bu.a 4
15.e even 4 1 3600.1.cc.a 2
15.e even 4 1 3600.1.cc.b 2
20.d odd 2 1 CM 2160.1.bu.a 4
24.f even 2 1 2880.1.bu.c 4
24.h odd 2 1 2880.1.bu.c 4
36.f odd 6 1 inner 2160.1.bu.a 4
36.h even 6 1 720.1.bu.a 4
45.h odd 6 1 720.1.bu.a 4
45.j even 6 1 inner 2160.1.bu.a 4
45.l even 12 1 3600.1.cc.a 2
45.l even 12 1 3600.1.cc.b 2
60.h even 2 1 720.1.bu.a 4
60.l odd 4 1 3600.1.cc.a 2
60.l odd 4 1 3600.1.cc.b 2
72.j odd 6 1 2880.1.bu.c 4
72.l even 6 1 2880.1.bu.c 4
120.i odd 2 1 2880.1.bu.c 4
120.m even 2 1 2880.1.bu.c 4
180.n even 6 1 720.1.bu.a 4
180.p odd 6 1 inner 2160.1.bu.a 4
180.v odd 12 1 3600.1.cc.a 2
180.v odd 12 1 3600.1.cc.b 2
360.bd even 6 1 2880.1.bu.c 4
360.bh odd 6 1 2880.1.bu.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
720.1.bu.a 4 3.b odd 2 1
720.1.bu.a 4 9.d odd 6 1
720.1.bu.a 4 12.b even 2 1
720.1.bu.a 4 15.d odd 2 1
720.1.bu.a 4 36.h even 6 1
720.1.bu.a 4 45.h odd 6 1
720.1.bu.a 4 60.h even 2 1
720.1.bu.a 4 180.n even 6 1
2160.1.bu.a 4 1.a even 1 1 trivial
2160.1.bu.a 4 4.b odd 2 1 inner
2160.1.bu.a 4 5.b even 2 1 inner
2160.1.bu.a 4 9.c even 3 1 inner
2160.1.bu.a 4 20.d odd 2 1 CM
2160.1.bu.a 4 36.f odd 6 1 inner
2160.1.bu.a 4 45.j even 6 1 inner
2160.1.bu.a 4 180.p odd 6 1 inner
2880.1.bu.c 4 24.f even 2 1
2880.1.bu.c 4 24.h odd 2 1
2880.1.bu.c 4 72.j odd 6 1
2880.1.bu.c 4 72.l even 6 1
2880.1.bu.c 4 120.i odd 2 1
2880.1.bu.c 4 120.m even 2 1
2880.1.bu.c 4 360.bd even 6 1
2880.1.bu.c 4 360.bh odd 6 1
3600.1.cc.a 2 15.e even 4 1
3600.1.cc.a 2 45.l even 12 1
3600.1.cc.a 2 60.l odd 4 1
3600.1.cc.a 2 180.v odd 12 1
3600.1.cc.b 2 15.e even 4 1
3600.1.cc.b 2 45.l even 12 1
3600.1.cc.b 2 60.l odd 4 1
3600.1.cc.b 2 180.v odd 12 1

Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(2160, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$(T^{2} + T + 1)^{2}$$
$7$ $$T^{4} + 3T^{2} + 9$$
$11$ $$T^{4}$$
$13$ $$T^{4}$$
$17$ $$T^{4}$$
$19$ $$T^{4}$$
$23$ $$T^{4} + 3T^{2} + 9$$
$29$ $$(T^{2} + T + 1)^{2}$$
$31$ $$T^{4}$$
$37$ $$T^{4}$$
$41$ $$(T^{2} - T + 1)^{2}$$
$43$ $$T^{4}$$
$47$ $$T^{4} + 3T^{2} + 9$$
$53$ $$T^{4}$$
$59$ $$T^{4}$$
$61$ $$(T^{2} + T + 1)^{2}$$
$67$ $$T^{4} + 3T^{2} + 9$$
$71$ $$T^{4}$$
$73$ $$T^{4}$$
$79$ $$T^{4}$$
$83$ $$T^{4} + 3T^{2} + 9$$
$89$ $$(T - 1)^{4}$$
$97$ $$T^{4}$$