# Properties

 Label 8640.2.a.bf 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 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)

 $$f(q)$$ $$=$$ $$q + q^{5} - 4 q^{7}+O(q^{10})$$ q + q^5 - 4 * q^7 $$q + q^{5} - 4 q^{7} - 2 q^{11} - 4 q^{13} + q^{17} + 5 q^{19} + 5 q^{23} + q^{25} - 8 q^{29} + 7 q^{31} - 4 q^{35} + 6 q^{37} + 6 q^{41} + 2 q^{43} + 8 q^{47} + 9 q^{49} - 9 q^{53} - 2 q^{55} - 4 q^{59} - 13 q^{61} - 4 q^{65} + 10 q^{67} - 6 q^{71} - 6 q^{73} + 8 q^{77} + 9 q^{79} + 17 q^{83} + q^{85} - 6 q^{89} + 16 q^{91} + 5 q^{95} - 8 q^{97}+O(q^{100})$$ q + q^5 - 4 * q^7 - 2 * q^11 - 4 * q^13 + q^17 + 5 * q^19 + 5 * q^23 + q^25 - 8 * q^29 + 7 * q^31 - 4 * q^35 + 6 * q^37 + 6 * q^41 + 2 * q^43 + 8 * q^47 + 9 * q^49 - 9 * q^53 - 2 * q^55 - 4 * q^59 - 13 * q^61 - 4 * q^65 + 10 * q^67 - 6 * q^71 - 6 * q^73 + 8 * q^77 + 9 * q^79 + 17 * q^83 + q^85 - 6 * q^89 + 16 * q^91 + 5 * q^95 - 8 * 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 −4.00000 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.bf 1
3.b odd 2 1 8640.2.a.a 1
4.b odd 2 1 8640.2.a.cg 1
8.b even 2 1 1080.2.a.a 1
8.d odd 2 1 2160.2.a.l 1
12.b even 2 1 8640.2.a.bd 1
24.f even 2 1 2160.2.a.w 1
24.h odd 2 1 1080.2.a.g yes 1
40.f even 2 1 5400.2.a.bu 1
40.i odd 4 2 5400.2.f.s 2
72.j odd 6 2 3240.2.q.l 2
72.n even 6 2 3240.2.q.w 2
120.i odd 2 1 5400.2.a.br 1
120.w even 4 2 5400.2.f.l 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1080.2.a.a 1 8.b even 2 1
1080.2.a.g yes 1 24.h odd 2 1
2160.2.a.l 1 8.d odd 2 1
2160.2.a.w 1 24.f even 2 1
3240.2.q.l 2 72.j odd 6 2
3240.2.q.w 2 72.n even 6 2
5400.2.a.br 1 120.i odd 2 1
5400.2.a.bu 1 40.f even 2 1
5400.2.f.l 2 120.w even 4 2
5400.2.f.s 2 40.i odd 4 2
8640.2.a.a 1 3.b odd 2 1
8640.2.a.bd 1 12.b even 2 1
8640.2.a.bf 1 1.a even 1 1 trivial
8640.2.a.cg 1 4.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8640))$$:

 $$T_{7} + 4$$ T7 + 4 $$T_{11} + 2$$ T11 + 2 $$T_{13} + 4$$ T13 + 4 $$T_{17} - 1$$ T17 - 1 $$T_{19} - 5$$ T19 - 5

## Hecke characteristic polynomials

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