# Properties

 Label 8640.2.a.t Level $8640$ Weight $2$ Character orbit 8640.a Self dual yes Analytic conductor $68.991$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8640,2,Mod(1,8640)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8640, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8640.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$8640 = 2^{6} \cdot 3^{3} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8640.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: $$68.9907473464$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1080) 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)

 $$f(q)$$ $$=$$ $$q - q^{5} + 2 q^{7}+O(q^{10})$$ q - q^5 + 2 * q^7 $$q - q^{5} + 2 q^{7} - 4 q^{11} + 2 q^{13} + 5 q^{17} + 5 q^{19} + q^{23} + q^{25} + 2 q^{29} + 7 q^{31} - 2 q^{35} + 6 q^{37} - 4 q^{43} + 4 q^{47} - 3 q^{49} - 9 q^{53} + 4 q^{55} - 14 q^{59} + 11 q^{61} - 2 q^{65} - 14 q^{67} - 12 q^{73} - 8 q^{77} - 3 q^{79} + q^{83} - 5 q^{85} + 4 q^{91} - 5 q^{95} + 16 q^{97}+O(q^{100})$$ q - q^5 + 2 * q^7 - 4 * q^11 + 2 * q^13 + 5 * q^17 + 5 * q^19 + q^23 + q^25 + 2 * q^29 + 7 * q^31 - 2 * q^35 + 6 * q^37 - 4 * q^43 + 4 * q^47 - 3 * q^49 - 9 * q^53 + 4 * q^55 - 14 * q^59 + 11 * q^61 - 2 * q^65 - 14 * q^67 - 12 * q^73 - 8 * q^77 - 3 * q^79 + q^83 - 5 * q^85 + 4 * q^91 - 5 * q^95 + 16 * 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
 0
0 0 0 −1.00000 0 2.00000 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
$$2$$ $$+1$$
$$3$$ $$-1$$
$$5$$ $$+1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8640.2.a.t 1
3.b odd 2 1 8640.2.a.cd 1
4.b odd 2 1 8640.2.a.k 1
8.b even 2 1 1080.2.a.l yes 1
8.d odd 2 1 2160.2.a.m 1
12.b even 2 1 8640.2.a.bi 1
24.f even 2 1 2160.2.a.e 1
24.h odd 2 1 1080.2.a.e 1
40.f even 2 1 5400.2.a.q 1
40.i odd 4 2 5400.2.f.x 2
72.j odd 6 2 3240.2.q.p 2
72.n even 6 2 3240.2.q.b 2
120.i odd 2 1 5400.2.a.j 1
120.w even 4 2 5400.2.f.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1080.2.a.e 1 24.h odd 2 1
1080.2.a.l yes 1 8.b even 2 1
2160.2.a.e 1 24.f even 2 1
2160.2.a.m 1 8.d odd 2 1
3240.2.q.b 2 72.n even 6 2
3240.2.q.p 2 72.j odd 6 2
5400.2.a.j 1 120.i odd 2 1
5400.2.a.q 1 40.f even 2 1
5400.2.f.f 2 120.w even 4 2
5400.2.f.x 2 40.i odd 4 2
8640.2.a.k 1 4.b odd 2 1
8640.2.a.t 1 1.a even 1 1 trivial
8640.2.a.bi 1 12.b even 2 1
8640.2.a.cd 1 3.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(8640))$$:

 $$T_{7} - 2$$ T7 - 2 $$T_{11} + 4$$ T11 + 4 $$T_{13} - 2$$ T13 - 2 $$T_{17} - 5$$ T17 - 5 $$T_{19} - 5$$ T19 - 5

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T + 1$$
$7$ $$T - 2$$
$11$ $$T + 4$$
$13$ $$T - 2$$
$17$ $$T - 5$$
$19$ $$T - 5$$
$23$ $$T - 1$$
$29$ $$T - 2$$
$31$ $$T - 7$$
$37$ $$T - 6$$
$41$ $$T$$
$43$ $$T + 4$$
$47$ $$T - 4$$
$53$ $$T + 9$$
$59$ $$T + 14$$
$61$ $$T - 11$$
$67$ $$T + 14$$
$71$ $$T$$
$73$ $$T + 12$$
$79$ $$T + 3$$
$83$ $$T - 1$$
$89$ $$T$$
$97$ $$T - 16$$