# Properties

 Label 8640.2.a.df Level $8640$ Weight $2$ Character orbit 8640.a Self dual yes Analytic conductor $68.991$ Analytic rank $0$ Dimension $2$ 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: $$2$$ 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: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 135) 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.

 $$f(q)$$ $$=$$ $$q + q^{5} + (\beta + 1) q^{7}+O(q^{10})$$ q + q^5 + (b + 1) * q^7 $$q + q^{5} + (\beta + 1) q^{7} + ( - \beta + 1) q^{11} + (\beta - 3) q^{13} + (\beta + 2) q^{17} + \beta q^{19} + 3 q^{23} + q^{25} + (\beta + 5) q^{29} + (\beta - 2) q^{31} + (\beta + 1) q^{35} - 2 q^{37} + ( - \beta + 1) q^{41} + (\beta + 3) q^{43} + ( - 2 \beta + 2) q^{47} + (2 \beta + 7) q^{49} + ( - \beta - 2) q^{53} + ( - \beta + 1) q^{55} + ( - \beta - 5) q^{59} + ( - 2 \beta - 3) q^{61} + (\beta - 3) q^{65} + (2 \beta + 8) q^{67} + ( - \beta - 11) q^{71} + ( - \beta + 9) q^{73} - 12 q^{77} + (\beta - 8) q^{79} + 3 q^{83} + (\beta + 2) q^{85} + ( - 3 \beta + 3) q^{89} + ( - 2 \beta + 10) q^{91} + \beta q^{95} + 8 q^{97}+O(q^{100})$$ q + q^5 + (b + 1) * q^7 + (-b + 1) * q^11 + (b - 3) * q^13 + (b + 2) * q^17 + b * q^19 + 3 * q^23 + q^25 + (b + 5) * q^29 + (b - 2) * q^31 + (b + 1) * q^35 - 2 * q^37 + (-b + 1) * q^41 + (b + 3) * q^43 + (-2*b + 2) * q^47 + (2*b + 7) * q^49 + (-b - 2) * q^53 + (-b + 1) * q^55 + (-b - 5) * q^59 + (-2*b - 3) * q^61 + (b - 3) * q^65 + (2*b + 8) * q^67 + (-b - 11) * q^71 + (-b + 9) * q^73 - 12 * q^77 + (b - 8) * q^79 + 3 * q^83 + (b + 2) * q^85 + (-3*b + 3) * q^89 + (-2*b + 10) * q^91 + b * q^95 + 8 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{5} + 2 q^{7}+O(q^{10})$$ 2 * q + 2 * q^5 + 2 * q^7 $$2 q + 2 q^{5} + 2 q^{7} + 2 q^{11} - 6 q^{13} + 4 q^{17} + 6 q^{23} + 2 q^{25} + 10 q^{29} - 4 q^{31} + 2 q^{35} - 4 q^{37} + 2 q^{41} + 6 q^{43} + 4 q^{47} + 14 q^{49} - 4 q^{53} + 2 q^{55} - 10 q^{59} - 6 q^{61} - 6 q^{65} + 16 q^{67} - 22 q^{71} + 18 q^{73} - 24 q^{77} - 16 q^{79} + 6 q^{83} + 4 q^{85} + 6 q^{89} + 20 q^{91} + 16 q^{97}+O(q^{100})$$ 2 * q + 2 * q^5 + 2 * q^7 + 2 * q^11 - 6 * q^13 + 4 * q^17 + 6 * q^23 + 2 * q^25 + 10 * q^29 - 4 * q^31 + 2 * q^35 - 4 * q^37 + 2 * q^41 + 6 * q^43 + 4 * q^47 + 14 * q^49 - 4 * q^53 + 2 * q^55 - 10 * q^59 - 6 * q^61 - 6 * q^65 + 16 * q^67 - 22 * q^71 + 18 * q^73 - 24 * q^77 - 16 * q^79 + 6 * q^83 + 4 * q^85 + 6 * q^89 + 20 * q^91 + 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
 −1.30278 2.30278
0 0 0 1.00000 0 −2.60555 0 0 0
1.2 0 0 0 1.00000 0 4.60555 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
$$2$$ $$1$$
$$3$$ $$1$$
$$5$$ $$-1$$

## 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.df 2
3.b odd 2 1 8640.2.a.cr 2
4.b odd 2 1 8640.2.a.cy 2
8.b even 2 1 135.2.a.d yes 2
8.d odd 2 1 2160.2.a.y 2
12.b even 2 1 8640.2.a.ck 2
24.f even 2 1 2160.2.a.ba 2
24.h odd 2 1 135.2.a.c 2
40.f even 2 1 675.2.a.k 2
40.i odd 4 2 675.2.b.h 4
56.h odd 2 1 6615.2.a.v 2
72.j odd 6 2 405.2.e.k 4
72.n even 6 2 405.2.e.j 4
120.i odd 2 1 675.2.a.p 2
120.w even 4 2 675.2.b.i 4
168.i even 2 1 6615.2.a.p 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.2.a.c 2 24.h odd 2 1
135.2.a.d yes 2 8.b even 2 1
405.2.e.j 4 72.n even 6 2
405.2.e.k 4 72.j odd 6 2
675.2.a.k 2 40.f even 2 1
675.2.a.p 2 120.i odd 2 1
675.2.b.h 4 40.i odd 4 2
675.2.b.i 4 120.w even 4 2
2160.2.a.y 2 8.d odd 2 1
2160.2.a.ba 2 24.f even 2 1
6615.2.a.p 2 168.i even 2 1
6615.2.a.v 2 56.h odd 2 1
8640.2.a.ck 2 12.b even 2 1
8640.2.a.cr 2 3.b odd 2 1
8640.2.a.cy 2 4.b odd 2 1
8640.2.a.df 2 1.a even 1 1 trivial

## Hecke kernels

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

 $$T_{7}^{2} - 2T_{7} - 12$$ T7^2 - 2*T7 - 12 $$T_{11}^{2} - 2T_{11} - 12$$ T11^2 - 2*T11 - 12 $$T_{13}^{2} + 6T_{13} - 4$$ T13^2 + 6*T13 - 4 $$T_{17}^{2} - 4T_{17} - 9$$ T17^2 - 4*T17 - 9 $$T_{19}^{2} - 13$$ T19^2 - 13

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$(T - 1)^{2}$$
$7$ $$T^{2} - 2T - 12$$
$11$ $$T^{2} - 2T - 12$$
$13$ $$T^{2} + 6T - 4$$
$17$ $$T^{2} - 4T - 9$$
$19$ $$T^{2} - 13$$
$23$ $$(T - 3)^{2}$$
$29$ $$T^{2} - 10T + 12$$
$31$ $$T^{2} + 4T - 9$$
$37$ $$(T + 2)^{2}$$
$41$ $$T^{2} - 2T - 12$$
$43$ $$T^{2} - 6T - 4$$
$47$ $$T^{2} - 4T - 48$$
$53$ $$T^{2} + 4T - 9$$
$59$ $$T^{2} + 10T + 12$$
$61$ $$T^{2} + 6T - 43$$
$67$ $$T^{2} - 16T + 12$$
$71$ $$T^{2} + 22T + 108$$
$73$ $$T^{2} - 18T + 68$$
$79$ $$T^{2} + 16T + 51$$
$83$ $$(T - 3)^{2}$$
$89$ $$T^{2} - 6T - 108$$
$97$ $$(T - 8)^{2}$$