# Properties

 Label 864.1.o.a Level $864$ Weight $1$ Character orbit 864.o Analytic conductor $0.431$ Analytic rank $0$ Dimension $4$ Projective image $A_{4}$ CM/RM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [864,1,Mod(127,864)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(864, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("864.127");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$864 = 2^{5} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 864.o (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: $$0.431192170915$$ 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 288) Projective image: $$A_{4}$$ Projective field: Galois closure of 4.0.5184.1

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \zeta_{12}^{4} q^{5} + \zeta_{12}^{5} q^{7}+O(q^{10})$$ q + z^4 * q^5 + z^5 * q^7 $$q + \zeta_{12}^{4} q^{5} + \zeta_{12}^{5} q^{7} + \zeta_{12}^{5} q^{11} + \zeta_{12}^{4} q^{13} - \zeta_{12} q^{23} + \zeta_{12}^{2} q^{29} + \zeta_{12} q^{31} - \zeta_{12}^{3} q^{35} - \zeta_{12}^{4} q^{41} + \zeta_{12}^{5} q^{43} - \zeta_{12}^{5} q^{47} - \zeta_{12}^{3} q^{55} + \zeta_{12} q^{59} - \zeta_{12}^{2} q^{61} - \zeta_{12}^{2} q^{65} + \zeta_{12} q^{67} + \zeta_{12}^{3} q^{71} - \zeta_{12}^{4} q^{77} - \zeta_{12}^{5} q^{79} - \zeta_{12}^{5} q^{83} - \zeta_{12}^{3} q^{91} - \zeta_{12}^{2} q^{97} +O(q^{100})$$ q + z^4 * q^5 + z^5 * q^7 + z^5 * q^11 + z^4 * q^13 - z * q^23 + z^2 * q^29 + z * q^31 - z^3 * q^35 - z^4 * q^41 + z^5 * q^43 - z^5 * q^47 - z^3 * q^55 + z * q^59 - z^2 * q^61 - z^2 * q^65 + z * q^67 + z^3 * q^71 - z^4 * q^77 - z^5 * q^79 - z^5 * q^83 - z^3 * q^91 - z^2 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{5}+O(q^{10})$$ 4 * q - 2 * q^5 $$4 q - 2 q^{5} - 2 q^{13} + 2 q^{29} + 2 q^{41} - 2 q^{61} - 2 q^{65} + 2 q^{77} - 2 q^{97}+O(q^{100})$$ 4 * q - 2 * q^5 - 2 * q^13 + 2 * q^29 + 2 * q^41 - 2 * q^61 - 2 * q^65 + 2 * q^77 - 2 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/864\mathbb{Z}\right)^\times$$.

 $$n$$ $$325$$ $$353$$ $$703$$ $$\chi(n)$$ $$1$$ $$\zeta_{12}^{4}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
127.1
 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 0.500000i 0 0 0
127.2 0 0 0 −0.500000 0.866025i 0 0.866025 + 0.500000i 0 0 0
415.1 0 0 0 −0.500000 + 0.866025i 0 −0.866025 + 0.500000i 0 0 0
415.2 0 0 0 −0.500000 + 0.866025i 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
4.b odd 2 1 inner
9.c even 3 1 inner
36.f odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 864.1.o.a 4
3.b odd 2 1 288.1.o.a 4
4.b odd 2 1 inner 864.1.o.a 4
8.b even 2 1 1728.1.o.a 4
8.d odd 2 1 1728.1.o.a 4
9.c even 3 1 inner 864.1.o.a 4
9.c even 3 1 2592.1.g.b 2
9.d odd 6 1 288.1.o.a 4
9.d odd 6 1 2592.1.g.a 2
12.b even 2 1 288.1.o.a 4
24.f even 2 1 576.1.o.a 4
24.h odd 2 1 576.1.o.a 4
36.f odd 6 1 inner 864.1.o.a 4
36.f odd 6 1 2592.1.g.b 2
36.h even 6 1 288.1.o.a 4
36.h even 6 1 2592.1.g.a 2
48.i odd 4 1 2304.1.t.a 4
48.i odd 4 1 2304.1.t.b 4
48.k even 4 1 2304.1.t.a 4
48.k even 4 1 2304.1.t.b 4
72.j odd 6 1 576.1.o.a 4
72.l even 6 1 576.1.o.a 4
72.n even 6 1 1728.1.o.a 4
72.p odd 6 1 1728.1.o.a 4
144.u even 12 1 2304.1.t.a 4
144.u even 12 1 2304.1.t.b 4
144.w odd 12 1 2304.1.t.a 4
144.w odd 12 1 2304.1.t.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
288.1.o.a 4 3.b odd 2 1
288.1.o.a 4 9.d odd 6 1
288.1.o.a 4 12.b even 2 1
288.1.o.a 4 36.h even 6 1
576.1.o.a 4 24.f even 2 1
576.1.o.a 4 24.h odd 2 1
576.1.o.a 4 72.j odd 6 1
576.1.o.a 4 72.l even 6 1
864.1.o.a 4 1.a even 1 1 trivial
864.1.o.a 4 4.b odd 2 1 inner
864.1.o.a 4 9.c even 3 1 inner
864.1.o.a 4 36.f odd 6 1 inner
1728.1.o.a 4 8.b even 2 1
1728.1.o.a 4 8.d odd 2 1
1728.1.o.a 4 72.n even 6 1
1728.1.o.a 4 72.p odd 6 1
2304.1.t.a 4 48.i odd 4 1
2304.1.t.a 4 48.k even 4 1
2304.1.t.a 4 144.u even 12 1
2304.1.t.a 4 144.w odd 12 1
2304.1.t.b 4 48.i odd 4 1
2304.1.t.b 4 48.k even 4 1
2304.1.t.b 4 144.u even 12 1
2304.1.t.b 4 144.w odd 12 1
2592.1.g.a 2 9.d odd 6 1
2592.1.g.a 2 36.h even 6 1
2592.1.g.b 2 9.c even 3 1
2592.1.g.b 2 36.f odd 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(864, [\chi])$$.

## Hecke characteristic polynomials

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