# Properties

 Label 8640.2.a.s Level $8640$ Weight $2$ Character orbit 8640.a Self dual yes Analytic conductor $68.991$ Analytic rank $1$ 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: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 540) 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)

 $$f(q)$$ $$=$$ $$q - q^{5} + q^{7}+O(q^{10})$$ q - q^5 + q^7 $$q - q^{5} + q^{7} + 6 q^{11} + q^{13} - q^{19} - 6 q^{23} + q^{25} - 6 q^{29} - 8 q^{31} - q^{35} + 7 q^{37} - 6 q^{41} - 4 q^{43} - 12 q^{47} - 6 q^{49} + 6 q^{53} - 6 q^{55} - 11 q^{61} - q^{65} - 7 q^{67} + 6 q^{71} + 11 q^{73} + 6 q^{77} + q^{79} + 6 q^{83} - 12 q^{89} + q^{91} + q^{95} - 13 q^{97}+O(q^{100})$$ q - q^5 + q^7 + 6 * q^11 + q^13 - q^19 - 6 * q^23 + q^25 - 6 * q^29 - 8 * q^31 - q^35 + 7 * q^37 - 6 * q^41 - 4 * q^43 - 12 * q^47 - 6 * q^49 + 6 * q^53 - 6 * q^55 - 11 * q^61 - q^65 - 7 * q^67 + 6 * q^71 + 11 * q^73 + 6 * q^77 + q^79 + 6 * q^83 - 12 * q^89 + q^91 + q^95 - 13 * 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 1.00000 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.s 1
3.b odd 2 1 8640.2.a.bu 1
4.b odd 2 1 8640.2.a.l 1
8.b even 2 1 2160.2.a.s 1
8.d odd 2 1 540.2.a.e yes 1
12.b even 2 1 8640.2.a.br 1
24.f even 2 1 540.2.a.b 1
24.h odd 2 1 2160.2.a.h 1
40.e odd 2 1 2700.2.a.m 1
40.k even 4 2 2700.2.d.k 2
72.l even 6 2 1620.2.i.j 2
72.p odd 6 2 1620.2.i.d 2
120.m even 2 1 2700.2.a.k 1
120.q odd 4 2 2700.2.d.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
540.2.a.b 1 24.f even 2 1
540.2.a.e yes 1 8.d odd 2 1
1620.2.i.d 2 72.p odd 6 2
1620.2.i.j 2 72.l even 6 2
2160.2.a.h 1 24.h odd 2 1
2160.2.a.s 1 8.b even 2 1
2700.2.a.k 1 120.m even 2 1
2700.2.a.m 1 40.e odd 2 1
2700.2.d.a 2 120.q odd 4 2
2700.2.d.k 2 40.k even 4 2
8640.2.a.l 1 4.b odd 2 1
8640.2.a.s 1 1.a even 1 1 trivial
8640.2.a.br 1 12.b even 2 1
8640.2.a.bu 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8640))$$:

 $$T_{7} - 1$$ T7 - 1 $$T_{11} - 6$$ T11 - 6 $$T_{13} - 1$$ T13 - 1 $$T_{17}$$ T17 $$T_{19} + 1$$ T19 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T + 1$$
$7$ $$T - 1$$
$11$ $$T - 6$$
$13$ $$T - 1$$
$17$ $$T$$
$19$ $$T + 1$$
$23$ $$T + 6$$
$29$ $$T + 6$$
$31$ $$T + 8$$
$37$ $$T - 7$$
$41$ $$T + 6$$
$43$ $$T + 4$$
$47$ $$T + 12$$
$53$ $$T - 6$$
$59$ $$T$$
$61$ $$T + 11$$
$67$ $$T + 7$$
$71$ $$T - 6$$
$73$ $$T - 11$$
$79$ $$T - 1$$
$83$ $$T - 6$$
$89$ $$T + 12$$
$97$ $$T + 13$$