# Properties

 Label 864.2.i.b Level $864$ Weight $2$ Character orbit 864.i Analytic conductor $6.899$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [864,2,Mod(289,864)]

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([0, 0, 4]))

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

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

chi := DirichletCharacter("864.289");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$864 = 2^{5} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 864.i (of order $$3$$, 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: $$6.89907473464$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 288) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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 $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 4 \zeta_{6} q^{5} + ( - 2 \zeta_{6} + 2) q^{7}+O(q^{10})$$ q + 4*z * q^5 + (-2*z + 2) * q^7 $$q + 4 \zeta_{6} q^{5} + ( - 2 \zeta_{6} + 2) q^{7} + (5 \zeta_{6} - 5) q^{11} + 2 \zeta_{6} q^{13} + 3 q^{17} + q^{19} - 6 \zeta_{6} q^{23} + (11 \zeta_{6} - 11) q^{25} + (2 \zeta_{6} - 2) q^{29} + 4 \zeta_{6} q^{31} + 8 q^{35} - 8 q^{37} + \zeta_{6} q^{41} + ( - 7 \zeta_{6} + 7) q^{43} + ( - 2 \zeta_{6} + 2) q^{47} + 3 \zeta_{6} q^{49} + 4 q^{53} - 20 q^{55} + 5 \zeta_{6} q^{59} + (8 \zeta_{6} - 8) q^{65} + 13 \zeta_{6} q^{67} - 8 q^{71} + 3 q^{73} + 10 \zeta_{6} q^{77} + (8 \zeta_{6} - 8) q^{79} + ( - 12 \zeta_{6} + 12) q^{83} + 12 \zeta_{6} q^{85} + 10 q^{89} + 4 q^{91} + 4 \zeta_{6} q^{95} + ( - 11 \zeta_{6} + 11) q^{97} +O(q^{100})$$ q + 4*z * q^5 + (-2*z + 2) * q^7 + (5*z - 5) * q^11 + 2*z * q^13 + 3 * q^17 + q^19 - 6*z * q^23 + (11*z - 11) * q^25 + (2*z - 2) * q^29 + 4*z * q^31 + 8 * q^35 - 8 * q^37 + z * q^41 + (-7*z + 7) * q^43 + (-2*z + 2) * q^47 + 3*z * q^49 + 4 * q^53 - 20 * q^55 + 5*z * q^59 + (8*z - 8) * q^65 + 13*z * q^67 - 8 * q^71 + 3 * q^73 + 10*z * q^77 + (8*z - 8) * q^79 + (-12*z + 12) * q^83 + 12*z * q^85 + 10 * q^89 + 4 * q^91 + 4*z * q^95 + (-11*z + 11) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{5} + 2 q^{7}+O(q^{10})$$ 2 * q + 4 * q^5 + 2 * q^7 $$2 q + 4 q^{5} + 2 q^{7} - 5 q^{11} + 2 q^{13} + 6 q^{17} + 2 q^{19} - 6 q^{23} - 11 q^{25} - 2 q^{29} + 4 q^{31} + 16 q^{35} - 16 q^{37} + q^{41} + 7 q^{43} + 2 q^{47} + 3 q^{49} + 8 q^{53} - 40 q^{55} + 5 q^{59} - 8 q^{65} + 13 q^{67} - 16 q^{71} + 6 q^{73} + 10 q^{77} - 8 q^{79} + 12 q^{83} + 12 q^{85} + 20 q^{89} + 8 q^{91} + 4 q^{95} + 11 q^{97}+O(q^{100})$$ 2 * q + 4 * q^5 + 2 * q^7 - 5 * q^11 + 2 * q^13 + 6 * q^17 + 2 * q^19 - 6 * q^23 - 11 * q^25 - 2 * q^29 + 4 * q^31 + 16 * q^35 - 16 * q^37 + q^41 + 7 * q^43 + 2 * q^47 + 3 * q^49 + 8 * q^53 - 40 * q^55 + 5 * q^59 - 8 * q^65 + 13 * q^67 - 16 * q^71 + 6 * q^73 + 10 * q^77 - 8 * q^79 + 12 * q^83 + 12 * q^85 + 20 * q^89 + 8 * q^91 + 4 * q^95 + 11 * q^97

## Character values

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

 $$n$$ $$325$$ $$353$$ $$703$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
289.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 0 0 2.00000 + 3.46410i 0 1.00000 1.73205i 0 0 0
577.1 0 0 0 2.00000 3.46410i 0 1.00000 + 1.73205i 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
9.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 864.2.i.b 2
3.b odd 2 1 288.2.i.b yes 2
4.b odd 2 1 864.2.i.a 2
8.b even 2 1 1728.2.i.b 2
8.d odd 2 1 1728.2.i.a 2
9.c even 3 1 inner 864.2.i.b 2
9.c even 3 1 2592.2.a.a 1
9.d odd 6 1 288.2.i.b yes 2
9.d odd 6 1 2592.2.a.g 1
12.b even 2 1 288.2.i.a 2
24.f even 2 1 576.2.i.h 2
24.h odd 2 1 576.2.i.b 2
36.f odd 6 1 864.2.i.a 2
36.f odd 6 1 2592.2.a.b 1
36.h even 6 1 288.2.i.a 2
36.h even 6 1 2592.2.a.h 1
72.j odd 6 1 576.2.i.b 2
72.j odd 6 1 5184.2.a.a 1
72.l even 6 1 576.2.i.h 2
72.l even 6 1 5184.2.a.b 1
72.n even 6 1 1728.2.i.b 2
72.n even 6 1 5184.2.a.be 1
72.p odd 6 1 1728.2.i.a 2
72.p odd 6 1 5184.2.a.bf 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
288.2.i.a 2 12.b even 2 1
288.2.i.a 2 36.h even 6 1
288.2.i.b yes 2 3.b odd 2 1
288.2.i.b yes 2 9.d odd 6 1
576.2.i.b 2 24.h odd 2 1
576.2.i.b 2 72.j odd 6 1
576.2.i.h 2 24.f even 2 1
576.2.i.h 2 72.l even 6 1
864.2.i.a 2 4.b odd 2 1
864.2.i.a 2 36.f odd 6 1
864.2.i.b 2 1.a even 1 1 trivial
864.2.i.b 2 9.c even 3 1 inner
1728.2.i.a 2 8.d odd 2 1
1728.2.i.a 2 72.p odd 6 1
1728.2.i.b 2 8.b even 2 1
1728.2.i.b 2 72.n even 6 1
2592.2.a.a 1 9.c even 3 1
2592.2.a.b 1 36.f odd 6 1
2592.2.a.g 1 9.d odd 6 1
2592.2.a.h 1 36.h even 6 1
5184.2.a.a 1 72.j odd 6 1
5184.2.a.b 1 72.l even 6 1
5184.2.a.be 1 72.n even 6 1
5184.2.a.bf 1 72.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 $$S_{2}^{\mathrm{new}}(864, [\chi])$$:

 $$T_{5}^{2} - 4T_{5} + 16$$ T5^2 - 4*T5 + 16 $$T_{7}^{2} - 2T_{7} + 4$$ T7^2 - 2*T7 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 4T + 16$$
$7$ $$T^{2} - 2T + 4$$
$11$ $$T^{2} + 5T + 25$$
$13$ $$T^{2} - 2T + 4$$
$17$ $$(T - 3)^{2}$$
$19$ $$(T - 1)^{2}$$
$23$ $$T^{2} + 6T + 36$$
$29$ $$T^{2} + 2T + 4$$
$31$ $$T^{2} - 4T + 16$$
$37$ $$(T + 8)^{2}$$
$41$ $$T^{2} - T + 1$$
$43$ $$T^{2} - 7T + 49$$
$47$ $$T^{2} - 2T + 4$$
$53$ $$(T - 4)^{2}$$
$59$ $$T^{2} - 5T + 25$$
$61$ $$T^{2}$$
$67$ $$T^{2} - 13T + 169$$
$71$ $$(T + 8)^{2}$$
$73$ $$(T - 3)^{2}$$
$79$ $$T^{2} + 8T + 64$$
$83$ $$T^{2} - 12T + 144$$
$89$ $$(T - 10)^{2}$$
$97$ $$T^{2} - 11T + 121$$