# Properties

 Label 2592.2.a.l Level $2592$ Weight $2$ Character orbit 2592.a Self dual yes Analytic conductor $20.697$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2592,2,Mod(1,2592)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2592, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2592.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2592 = 2^{5} \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2592.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: $$20.6972242039$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 288) 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 $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{5} + \beta q^{7}+O(q^{10})$$ q - q^5 + b * q^7 $$q - q^{5} + \beta q^{7} - \beta q^{11} - 3 q^{13} + 4 q^{17} - 4 \beta q^{19} + 5 \beta q^{23} - 4 q^{25} + q^{29} - 3 \beta q^{31} - \beta q^{35} - 8 q^{37} + 5 q^{41} + 5 \beta q^{43} - 7 \beta q^{47} - 4 q^{49} - 8 q^{53} + \beta q^{55} - \beta q^{59} - 7 q^{61} + 3 q^{65} + 5 \beta q^{67} - 2 \beta q^{71} - 12 q^{73} - 3 q^{77} + 3 \beta q^{79} + 5 \beta q^{83} - 4 q^{85} - 4 q^{89} - 3 \beta q^{91} + 4 \beta q^{95} - 3 q^{97} +O(q^{100})$$ q - q^5 + b * q^7 - b * q^11 - 3 * q^13 + 4 * q^17 - 4*b * q^19 + 5*b * q^23 - 4 * q^25 + q^29 - 3*b * q^31 - b * q^35 - 8 * q^37 + 5 * q^41 + 5*b * q^43 - 7*b * q^47 - 4 * q^49 - 8 * q^53 + b * q^55 - b * q^59 - 7 * q^61 + 3 * q^65 + 5*b * q^67 - 2*b * q^71 - 12 * q^73 - 3 * q^77 + 3*b * q^79 + 5*b * q^83 - 4 * q^85 - 4 * q^89 - 3*b * q^91 + 4*b * q^95 - 3 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{5}+O(q^{10})$$ 2 * q - 2 * q^5 $$2 q - 2 q^{5} - 6 q^{13} + 8 q^{17} - 8 q^{25} + 2 q^{29} - 16 q^{37} + 10 q^{41} - 8 q^{49} - 16 q^{53} - 14 q^{61} + 6 q^{65} - 24 q^{73} - 6 q^{77} - 8 q^{85} - 8 q^{89} - 6 q^{97}+O(q^{100})$$ 2 * q - 2 * q^5 - 6 * q^13 + 8 * q^17 - 8 * q^25 + 2 * q^29 - 16 * q^37 + 10 * q^41 - 8 * q^49 - 16 * q^53 - 14 * q^61 + 6 * q^65 - 24 * q^73 - 6 * q^77 - 8 * q^85 - 8 * q^89 - 6 * 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
 −1.73205 1.73205
0 0 0 −1.00000 0 −1.73205 0 0 0
1.2 0 0 0 −1.00000 0 1.73205 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
$$2$$ $$1$$
$$3$$ $$1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2592.2.a.l 2
3.b odd 2 1 2592.2.a.p 2
4.b odd 2 1 inner 2592.2.a.l 2
8.b even 2 1 5184.2.a.bx 2
8.d odd 2 1 5184.2.a.bx 2
9.c even 3 2 288.2.i.d 4
9.d odd 6 2 864.2.i.d 4
12.b even 2 1 2592.2.a.p 2
24.f even 2 1 5184.2.a.bl 2
24.h odd 2 1 5184.2.a.bl 2
36.f odd 6 2 288.2.i.d 4
36.h even 6 2 864.2.i.d 4
72.j odd 6 2 1728.2.i.l 4
72.l even 6 2 1728.2.i.l 4
72.n even 6 2 576.2.i.k 4
72.p odd 6 2 576.2.i.k 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
288.2.i.d 4 9.c even 3 2
288.2.i.d 4 36.f odd 6 2
576.2.i.k 4 72.n even 6 2
576.2.i.k 4 72.p odd 6 2
864.2.i.d 4 9.d odd 6 2
864.2.i.d 4 36.h even 6 2
1728.2.i.l 4 72.j odd 6 2
1728.2.i.l 4 72.l even 6 2
2592.2.a.l 2 1.a even 1 1 trivial
2592.2.a.l 2 4.b odd 2 1 inner
2592.2.a.p 2 3.b odd 2 1
2592.2.a.p 2 12.b even 2 1
5184.2.a.bl 2 24.f even 2 1
5184.2.a.bl 2 24.h odd 2 1
5184.2.a.bx 2 8.b even 2 1
5184.2.a.bx 2 8.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2592))$$:

 $$T_{5} + 1$$ T5 + 1 $$T_{7}^{2} - 3$$ T7^2 - 3 $$T_{11}^{2} - 3$$ T11^2 - 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$(T + 1)^{2}$$
$7$ $$T^{2} - 3$$
$11$ $$T^{2} - 3$$
$13$ $$(T + 3)^{2}$$
$17$ $$(T - 4)^{2}$$
$19$ $$T^{2} - 48$$
$23$ $$T^{2} - 75$$
$29$ $$(T - 1)^{2}$$
$31$ $$T^{2} - 27$$
$37$ $$(T + 8)^{2}$$
$41$ $$(T - 5)^{2}$$
$43$ $$T^{2} - 75$$
$47$ $$T^{2} - 147$$
$53$ $$(T + 8)^{2}$$
$59$ $$T^{2} - 3$$
$61$ $$(T + 7)^{2}$$
$67$ $$T^{2} - 75$$
$71$ $$T^{2} - 12$$
$73$ $$(T + 12)^{2}$$
$79$ $$T^{2} - 27$$
$83$ $$T^{2} - 75$$
$89$ $$(T + 4)^{2}$$
$97$ $$(T + 3)^{2}$$