# Properties

 Label 2592.2.a.d Level $2592$ Weight $2$ Character orbit 2592.a Self dual yes Analytic conductor $20.697$ 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] = [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: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes 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 $$q - q^{5} + 2 q^{7} - 2 q^{11} + q^{13} - 3 q^{17} + 2 q^{19} - 6 q^{23} - 4 q^{25} - q^{29} - 8 q^{31} - 2 q^{35} + q^{37} + 2 q^{41} + 10 q^{43} - 4 q^{47} - 3 q^{49} + 10 q^{53} + 2 q^{55} - 4 q^{59} + 9 q^{61} - q^{65} - 14 q^{67} - 10 q^{71} - 9 q^{73} - 4 q^{77} + 10 q^{79} - 12 q^{83} + 3 q^{85} - 11 q^{89} + 2 q^{91} - 2 q^{95} - 2 q^{97}+O(q^{100})$$ q - q^5 + 2 * q^7 - 2 * q^11 + q^13 - 3 * q^17 + 2 * q^19 - 6 * q^23 - 4 * q^25 - q^29 - 8 * q^31 - 2 * q^35 + q^37 + 2 * q^41 + 10 * q^43 - 4 * q^47 - 3 * q^49 + 10 * q^53 + 2 * q^55 - 4 * q^59 + 9 * q^61 - q^65 - 14 * q^67 - 10 * q^71 - 9 * q^73 - 4 * q^77 + 10 * q^79 - 12 * q^83 + 3 * q^85 - 11 * q^89 + 2 * q^91 - 2 * q^95 - 2 * 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
$$2$$ $$+1$$
$$3$$ $$+1$$

## 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 2592.2.a.d yes 1
3.b odd 2 1 2592.2.a.f yes 1
4.b odd 2 1 2592.2.a.c 1
8.b even 2 1 5184.2.a.w 1
8.d odd 2 1 5184.2.a.t 1
9.c even 3 2 2592.2.i.n 2
9.d odd 6 2 2592.2.i.j 2
12.b even 2 1 2592.2.a.e yes 1
24.f even 2 1 5184.2.a.j 1
24.h odd 2 1 5184.2.a.m 1
36.f odd 6 2 2592.2.i.o 2
36.h even 6 2 2592.2.i.k 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2592.2.a.c 1 4.b odd 2 1
2592.2.a.d yes 1 1.a even 1 1 trivial
2592.2.a.e yes 1 12.b even 2 1
2592.2.a.f yes 1 3.b odd 2 1
2592.2.i.j 2 9.d odd 6 2
2592.2.i.k 2 36.h even 6 2
2592.2.i.n 2 9.c even 3 2
2592.2.i.o 2 36.f odd 6 2
5184.2.a.j 1 24.f even 2 1
5184.2.a.m 1 24.h odd 2 1
5184.2.a.t 1 8.d odd 2 1
5184.2.a.w 1 8.b even 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 $$T_{7} - 2$$ T7 - 2 $$T_{11} + 2$$ T11 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T + 1$$
$7$ $$T - 2$$
$11$ $$T + 2$$
$13$ $$T - 1$$
$17$ $$T + 3$$
$19$ $$T - 2$$
$23$ $$T + 6$$
$29$ $$T + 1$$
$31$ $$T + 8$$
$37$ $$T - 1$$
$41$ $$T - 2$$
$43$ $$T - 10$$
$47$ $$T + 4$$
$53$ $$T - 10$$
$59$ $$T + 4$$
$61$ $$T - 9$$
$67$ $$T + 14$$
$71$ $$T + 10$$
$73$ $$T + 9$$
$79$ $$T - 10$$
$83$ $$T + 12$$
$89$ $$T + 11$$
$97$ $$T + 2$$