# Properties

 Label 8640.2.a.x 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 $$q - q^{5} + 2 q^{7} + 6 q^{13} - 7 q^{17} + 7 q^{19} + 7 q^{23} + q^{25} + 6 q^{29} - 3 q^{31} - 2 q^{35} + 6 q^{37} - 4 q^{41} + 8 q^{43} - 4 q^{47} - 3 q^{49} - 5 q^{53} - 6 q^{59} + 3 q^{61} - 6 q^{65} - 10 q^{67} + 12 q^{71} + 16 q^{73} - q^{79} - 9 q^{83} + 7 q^{85} + 4 q^{89} + 12 q^{91} - 7 q^{95} - 16 q^{97}+O(q^{100})$$ q - q^5 + 2 * q^7 + 6 * q^13 - 7 * q^17 + 7 * q^19 + 7 * q^23 + q^25 + 6 * q^29 - 3 * q^31 - 2 * q^35 + 6 * q^37 - 4 * q^41 + 8 * q^43 - 4 * q^47 - 3 * q^49 - 5 * q^53 - 6 * q^59 + 3 * q^61 - 6 * q^65 - 10 * q^67 + 12 * q^71 + 16 * q^73 - q^79 - 9 * q^83 + 7 * q^85 + 4 * q^89 + 12 * q^91 - 7 * 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.x 1
3.b odd 2 1 8640.2.a.ca 1
4.b odd 2 1 8640.2.a.h 1
8.b even 2 1 2160.2.a.u 1
8.d odd 2 1 1080.2.a.h yes 1
12.b even 2 1 8640.2.a.bm 1
24.f even 2 1 1080.2.a.b 1
24.h odd 2 1 2160.2.a.i 1
40.e odd 2 1 5400.2.a.bi 1
40.k even 4 2 5400.2.f.o 2
72.l even 6 2 3240.2.q.v 2
72.p odd 6 2 3240.2.q.i 2
120.m even 2 1 5400.2.a.bh 1
120.q odd 4 2 5400.2.f.n 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1080.2.a.b 1 24.f even 2 1
1080.2.a.h yes 1 8.d odd 2 1
2160.2.a.i 1 24.h odd 2 1
2160.2.a.u 1 8.b even 2 1
3240.2.q.i 2 72.p odd 6 2
3240.2.q.v 2 72.l even 6 2
5400.2.a.bh 1 120.m even 2 1
5400.2.a.bi 1 40.e odd 2 1
5400.2.f.n 2 120.q odd 4 2
5400.2.f.o 2 40.k even 4 2
8640.2.a.h 1 4.b odd 2 1
8640.2.a.x 1 1.a even 1 1 trivial
8640.2.a.bm 1 12.b even 2 1
8640.2.a.ca 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 $$T_{11}$$ T11 $$T_{13} - 6$$ T13 - 6 $$T_{17} + 7$$ T17 + 7 $$T_{19} - 7$$ T19 - 7

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T + 1$$
$7$ $$T - 2$$
$11$ $$T$$
$13$ $$T - 6$$
$17$ $$T + 7$$
$19$ $$T - 7$$
$23$ $$T - 7$$
$29$ $$T - 6$$
$31$ $$T + 3$$
$37$ $$T - 6$$
$41$ $$T + 4$$
$43$ $$T - 8$$
$47$ $$T + 4$$
$53$ $$T + 5$$
$59$ $$T + 6$$
$61$ $$T - 3$$
$67$ $$T + 10$$
$71$ $$T - 12$$
$73$ $$T - 16$$
$79$ $$T + 1$$
$83$ $$T + 9$$
$89$ $$T - 4$$
$97$ $$T + 16$$