# Properties

 Label 864.2.a.m Level $864$ Weight $2$ Character orbit 864.a Self dual yes Analytic conductor $6.899$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [864,2,Mod(1,864)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(864, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("864.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$864 = 2^{5} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 864.a (trivial)

## 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: yes Analytic conductor: $$6.89907473464$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{13})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 3$$ x^2 - x - 3 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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 $$\beta = \sqrt{13}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta q^{5} + \beta q^{7} +O(q^{10})$$ q - b * q^5 + b * q^7 $$q - \beta q^{5} + \beta q^{7} - q^{11} + 4 q^{13} - 2 \beta q^{19} + 6 q^{23} + 8 q^{25} + 2 \beta q^{29} - \beta q^{31} - 13 q^{35} + 10 q^{37} + 2 \beta q^{41} + 2 \beta q^{43} + 10 q^{47} + 6 q^{49} + \beta q^{53} + \beta q^{55} + 4 q^{59} - 4 \beta q^{65} + 2 \beta q^{67} - 8 q^{71} - 3 q^{73} - \beta q^{77} - 4 \beta q^{79} - 9 q^{83} - 2 \beta q^{89} + 4 \beta q^{91} + 26 q^{95} + 7 q^{97} +O(q^{100})$$ q - b * q^5 + b * q^7 - q^11 + 4 * q^13 - 2*b * q^19 + 6 * q^23 + 8 * q^25 + 2*b * q^29 - b * q^31 - 13 * q^35 + 10 * q^37 + 2*b * q^41 + 2*b * q^43 + 10 * q^47 + 6 * q^49 + b * q^53 + b * q^55 + 4 * q^59 - 4*b * q^65 + 2*b * q^67 - 8 * q^71 - 3 * q^73 - b * q^77 - 4*b * q^79 - 9 * q^83 - 2*b * q^89 + 4*b * q^91 + 26 * q^95 + 7 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q+O(q^{10})$$ 2 * q $$2 q - 2 q^{11} + 8 q^{13} + 12 q^{23} + 16 q^{25} - 26 q^{35} + 20 q^{37} + 20 q^{47} + 12 q^{49} + 8 q^{59} - 16 q^{71} - 6 q^{73} - 18 q^{83} + 52 q^{95} + 14 q^{97}+O(q^{100})$$ 2 * q - 2 * q^11 + 8 * q^13 + 12 * q^23 + 16 * q^25 - 26 * q^35 + 20 * q^37 + 20 * q^47 + 12 * q^49 + 8 * q^59 - 16 * q^71 - 6 * q^73 - 18 * q^83 + 52 * q^95 + 14 * q^97

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 2.30278 −1.30278
0 0 0 −3.60555 0 3.60555 0 0 0
1.2 0 0 0 3.60555 0 −3.60555 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
12.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 864.2.a.m 2
3.b odd 2 1 864.2.a.n yes 2
4.b odd 2 1 864.2.a.n yes 2
8.b even 2 1 1728.2.a.bd 2
8.d odd 2 1 1728.2.a.bc 2
9.c even 3 2 2592.2.i.bc 4
9.d odd 6 2 2592.2.i.bb 4
12.b even 2 1 inner 864.2.a.m 2
24.f even 2 1 1728.2.a.bd 2
24.h odd 2 1 1728.2.a.bc 2
36.f odd 6 2 2592.2.i.bb 4
36.h even 6 2 2592.2.i.bc 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
864.2.a.m 2 1.a even 1 1 trivial
864.2.a.m 2 12.b even 2 1 inner
864.2.a.n yes 2 3.b odd 2 1
864.2.a.n yes 2 4.b odd 2 1
1728.2.a.bc 2 8.d odd 2 1
1728.2.a.bc 2 24.h odd 2 1
1728.2.a.bd 2 8.b even 2 1
1728.2.a.bd 2 24.f even 2 1
2592.2.i.bb 4 9.d odd 6 2
2592.2.i.bb 4 36.f odd 6 2
2592.2.i.bc 4 9.c even 3 2
2592.2.i.bc 4 36.h even 6 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(864))$$:

 $$T_{5}^{2} - 13$$ T5^2 - 13 $$T_{7}^{2} - 13$$ T7^2 - 13 $$T_{11} + 1$$ T11 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 13$$
$7$ $$T^{2} - 13$$
$11$ $$(T + 1)^{2}$$
$13$ $$(T - 4)^{2}$$
$17$ $$T^{2}$$
$19$ $$T^{2} - 52$$
$23$ $$(T - 6)^{2}$$
$29$ $$T^{2} - 52$$
$31$ $$T^{2} - 13$$
$37$ $$(T - 10)^{2}$$
$41$ $$T^{2} - 52$$
$43$ $$T^{2} - 52$$
$47$ $$(T - 10)^{2}$$
$53$ $$T^{2} - 13$$
$59$ $$(T - 4)^{2}$$
$61$ $$T^{2}$$
$67$ $$T^{2} - 52$$
$71$ $$(T + 8)^{2}$$
$73$ $$(T + 3)^{2}$$
$79$ $$T^{2} - 208$$
$83$ $$(T + 9)^{2}$$
$89$ $$T^{2} - 52$$
$97$ $$(T - 7)^{2}$$
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