# Properties

 Label 5184.2.a.bf Level $5184$ Weight $2$ Character orbit 5184.a Self dual yes Analytic conductor $41.394$ 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] = [5184,2,Mod(1,5184)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5184, 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("5184.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$5184 = 2^{6} \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5184.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: $$41.3944484078$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ 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)

 $$f(q)$$ $$=$$ $$q + 4 q^{5} + 2 q^{7}+O(q^{10})$$ q + 4 * q^5 + 2 * q^7 $$q + 4 q^{5} + 2 q^{7} + 5 q^{11} + 2 q^{13} + 3 q^{17} + q^{19} - 6 q^{23} + 11 q^{25} - 2 q^{29} + 4 q^{31} + 8 q^{35} + 8 q^{37} - q^{41} - 7 q^{43} + 2 q^{47} - 3 q^{49} - 4 q^{53} + 20 q^{55} - 5 q^{59} + 8 q^{65} - 13 q^{67} + 8 q^{71} + 3 q^{73} + 10 q^{77} - 8 q^{79} - 12 q^{83} + 12 q^{85} + 10 q^{89} + 4 q^{91} + 4 q^{95} - 11 q^{97}+O(q^{100})$$ q + 4 * q^5 + 2 * q^7 + 5 * q^11 + 2 * q^13 + 3 * q^17 + q^19 - 6 * q^23 + 11 * q^25 - 2 * q^29 + 4 * q^31 + 8 * q^35 + 8 * q^37 - q^41 - 7 * q^43 + 2 * q^47 - 3 * q^49 - 4 * q^53 + 20 * q^55 - 5 * q^59 + 8 * q^65 - 13 * q^67 + 8 * q^71 + 3 * q^73 + 10 * q^77 - 8 * q^79 - 12 * q^83 + 12 * q^85 + 10 * q^89 + 4 * q^91 + 4 * q^95 - 11 * 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 4.00000 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 5184.2.a.bf 1
3.b odd 2 1 5184.2.a.b 1
4.b odd 2 1 5184.2.a.be 1
8.b even 2 1 2592.2.a.b 1
8.d odd 2 1 2592.2.a.a 1
9.c even 3 2 1728.2.i.a 2
9.d odd 6 2 576.2.i.h 2
12.b even 2 1 5184.2.a.a 1
24.f even 2 1 2592.2.a.g 1
24.h odd 2 1 2592.2.a.h 1
36.f odd 6 2 1728.2.i.b 2
36.h even 6 2 576.2.i.b 2
72.j odd 6 2 288.2.i.a 2
72.l even 6 2 288.2.i.b yes 2
72.n even 6 2 864.2.i.a 2
72.p odd 6 2 864.2.i.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
288.2.i.a 2 72.j odd 6 2
288.2.i.b yes 2 72.l even 6 2
576.2.i.b 2 36.h even 6 2
576.2.i.h 2 9.d odd 6 2
864.2.i.a 2 72.n even 6 2
864.2.i.b 2 72.p odd 6 2
1728.2.i.a 2 9.c even 3 2
1728.2.i.b 2 36.f odd 6 2
2592.2.a.a 1 8.d odd 2 1
2592.2.a.b 1 8.b even 2 1
2592.2.a.g 1 24.f even 2 1
2592.2.a.h 1 24.h odd 2 1
5184.2.a.a 1 12.b even 2 1
5184.2.a.b 1 3.b odd 2 1
5184.2.a.be 1 4.b odd 2 1
5184.2.a.bf 1 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(5184))$$:

 $$T_{5} - 4$$ T5 - 4 $$T_{7} - 2$$ T7 - 2 $$T_{11} - 5$$ T11 - 5

## Hecke characteristic polynomials

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