# Properties

 Label 1080.1.i.e Level $1080$ Weight $1$ Character orbit 1080.i Analytic conductor $0.539$ Analytic rank $0$ Dimension $4$ Projective image $D_{6}$ CM discriminant -24 Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1080,1,Mod(269,1080)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1080, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 1, 1]))

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

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

chi := DirichletCharacter("1080.269");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1080 = 2^{3} \cdot 3^{3} \cdot 5$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1080.i (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: $$0.538990213644$$ Analytic rank: $$0$$ Dimension: $$4$$ 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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{6}$$ Projective field: Galois closure of 6.2.46656000.2

## $q$-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.

 $$f(q)$$ $$=$$ $$q - \zeta_{12}^{3} q^{2} - q^{4} - \zeta_{12} q^{5} + (\zeta_{12}^{4} + \zeta_{12}^{2}) q^{7} + \zeta_{12}^{3} q^{8} +O(q^{10})$$ q - z^3 * q^2 - q^4 - z * q^5 + (z^4 + z^2) * q^7 + z^3 * q^8 $$q - \zeta_{12}^{3} q^{2} - q^{4} - \zeta_{12} q^{5} + (\zeta_{12}^{4} + \zeta_{12}^{2}) q^{7} + \zeta_{12}^{3} q^{8} + \zeta_{12}^{4} q^{10} + ( - \zeta_{12}^{5} + \zeta_{12}) q^{11} + ( - \zeta_{12}^{5} + \zeta_{12}) q^{14} + q^{16} + \zeta_{12} q^{20} + ( - \zeta_{12}^{4} - \zeta_{12}^{2}) q^{22} + \zeta_{12}^{2} q^{25} + ( - \zeta_{12}^{4} - \zeta_{12}^{2}) q^{28} + q^{31} - \zeta_{12}^{3} q^{32} + ( - \zeta_{12}^{5} - \zeta_{12}^{3}) q^{35} - \zeta_{12}^{4} q^{40} + (\zeta_{12}^{5} - \zeta_{12}) q^{44} + (\zeta_{12}^{4} - \zeta_{12}^{2} - 1) q^{49} - \zeta_{12}^{5} q^{50} + \zeta_{12}^{3} q^{53} + ( - \zeta_{12}^{2} - 1) q^{55} + (\zeta_{12}^{5} - \zeta_{12}) q^{56} - \zeta_{12}^{3} q^{62} - q^{64} + ( - \zeta_{12}^{2} - 1) q^{70} + (\zeta_{12}^{4} + \zeta_{12}^{2}) q^{73} + (\zeta_{12}^{5} + \zeta_{12}^{3} + \zeta_{12}) q^{77} + q^{79} - \zeta_{12} q^{80} - \zeta_{12}^{3} q^{83} + (\zeta_{12}^{4} + \zeta_{12}^{2}) q^{88} + ( - \zeta_{12}^{4} - \zeta_{12}^{2}) q^{97} + (\zeta_{12}^{5} + \zeta_{12}^{3} + \zeta_{12}) q^{98} +O(q^{100})$$ q - z^3 * q^2 - q^4 - z * q^5 + (z^4 + z^2) * q^7 + z^3 * q^8 + z^4 * q^10 + (-z^5 + z) * q^11 + (-z^5 + z) * q^14 + q^16 + z * q^20 + (-z^4 - z^2) * q^22 + z^2 * q^25 + (-z^4 - z^2) * q^28 + q^31 - z^3 * q^32 + (-z^5 - z^3) * q^35 - z^4 * q^40 + (z^5 - z) * q^44 + (z^4 - z^2 - 1) * q^49 - z^5 * q^50 + z^3 * q^53 + (-z^2 - 1) * q^55 + (z^5 - z) * q^56 - z^3 * q^62 - q^64 + (-z^2 - 1) * q^70 + (z^4 + z^2) * q^73 + (z^5 + z^3 + z) * q^77 + q^79 - z * q^80 - z^3 * q^83 + (z^4 + z^2) * q^88 + (-z^4 - z^2) * q^97 + (z^5 + z^3 + z) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{4}+O(q^{10})$$ 4 * q - 4 * q^4 $$4 q - 4 q^{4} - 2 q^{10} + 4 q^{16} + 2 q^{25} + 4 q^{31} + 2 q^{40} - 8 q^{49} - 6 q^{55} - 4 q^{64} - 6 q^{70} + 8 q^{79}+O(q^{100})$$ 4 * q - 4 * q^4 - 2 * q^10 + 4 * q^16 + 2 * q^25 + 4 * q^31 + 2 * q^40 - 8 * q^49 - 6 * q^55 - 4 * q^64 - 6 * q^70 + 8 * q^79

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1080\mathbb{Z}\right)^\times$$.

 $$n$$ $$217$$ $$271$$ $$541$$ $$1001$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
269.1
 0.866025 + 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 − 0.500000i
1.00000i 0 −1.00000 −0.866025 0.500000i 0 1.73205i 1.00000i 0 −0.500000 + 0.866025i
269.2 1.00000i 0 −1.00000 0.866025 0.500000i 0 1.73205i 1.00000i 0 −0.500000 0.866025i
269.3 1.00000i 0 −1.00000 −0.866025 + 0.500000i 0 1.73205i 1.00000i 0 −0.500000 0.866025i
269.4 1.00000i 0 −1.00000 0.866025 + 0.500000i 0 1.73205i 1.00000i 0 −0.500000 + 0.866025i
 $$n$$: 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
24.h odd 2 1 CM by $$\Q(\sqrt{-6})$$
3.b odd 2 1 inner
5.b even 2 1 inner
8.b even 2 1 inner
15.d odd 2 1 inner
40.f even 2 1 inner
120.i odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1080.1.i.e 4
3.b odd 2 1 inner 1080.1.i.e 4
5.b even 2 1 inner 1080.1.i.e 4
8.b even 2 1 inner 1080.1.i.e 4
9.c even 3 1 3240.1.bh.f 4
9.c even 3 1 3240.1.bh.i 4
9.d odd 6 1 3240.1.bh.f 4
9.d odd 6 1 3240.1.bh.i 4
15.d odd 2 1 inner 1080.1.i.e 4
24.h odd 2 1 CM 1080.1.i.e 4
40.f even 2 1 inner 1080.1.i.e 4
45.h odd 6 1 3240.1.bh.f 4
45.h odd 6 1 3240.1.bh.i 4
45.j even 6 1 3240.1.bh.f 4
45.j even 6 1 3240.1.bh.i 4
72.j odd 6 1 3240.1.bh.f 4
72.j odd 6 1 3240.1.bh.i 4
72.n even 6 1 3240.1.bh.f 4
72.n even 6 1 3240.1.bh.i 4
120.i odd 2 1 inner 1080.1.i.e 4
360.bh odd 6 1 3240.1.bh.f 4
360.bh odd 6 1 3240.1.bh.i 4
360.bk even 6 1 3240.1.bh.f 4
360.bk even 6 1 3240.1.bh.i 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1080.1.i.e 4 1.a even 1 1 trivial
1080.1.i.e 4 3.b odd 2 1 inner
1080.1.i.e 4 5.b even 2 1 inner
1080.1.i.e 4 8.b even 2 1 inner
1080.1.i.e 4 15.d odd 2 1 inner
1080.1.i.e 4 24.h odd 2 1 CM
1080.1.i.e 4 40.f even 2 1 inner
1080.1.i.e 4 120.i odd 2 1 inner
3240.1.bh.f 4 9.c even 3 1
3240.1.bh.f 4 9.d odd 6 1
3240.1.bh.f 4 45.h odd 6 1
3240.1.bh.f 4 45.j even 6 1
3240.1.bh.f 4 72.j odd 6 1
3240.1.bh.f 4 72.n even 6 1
3240.1.bh.f 4 360.bh odd 6 1
3240.1.bh.f 4 360.bk even 6 1
3240.1.bh.i 4 9.c even 3 1
3240.1.bh.i 4 9.d odd 6 1
3240.1.bh.i 4 45.h odd 6 1
3240.1.bh.i 4 45.j even 6 1
3240.1.bh.i 4 72.j odd 6 1
3240.1.bh.i 4 72.n even 6 1
3240.1.bh.i 4 360.bh odd 6 1
3240.1.bh.i 4 360.bk even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(1080, [\chi])$$:

 $$T_{7}^{2} + 3$$ T7^2 + 3 $$T_{11}^{2} - 3$$ T11^2 - 3 $$T_{13}$$ T13

## Hecke characteristic polynomials

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