# Properties

 Label 1584.4.a.p Level $1584$ Weight $4$ Character orbit 1584.a Self dual yes Analytic conductor $93.459$ 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] = [1584,4,Mod(1,1584)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1584, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1584.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1584 = 2^{4} \cdot 3^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1584.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: $$93.4590254491$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 44) 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)

 $$f(q)$$ $$=$$ $$q + 7 q^{5} + 26 q^{7}+O(q^{10})$$ q + 7 * q^5 + 26 * q^7 $$q + 7 q^{5} + 26 q^{7} - 11 q^{11} + 52 q^{13} - 46 q^{17} + 96 q^{19} + 27 q^{23} - 76 q^{25} - 16 q^{29} + 293 q^{31} + 182 q^{35} - 29 q^{37} + 472 q^{41} + 110 q^{43} - 224 q^{47} + 333 q^{49} - 754 q^{53} - 77 q^{55} + 825 q^{59} - 548 q^{61} + 364 q^{65} + 123 q^{67} + 1001 q^{71} - 1020 q^{73} - 286 q^{77} - 526 q^{79} - 158 q^{83} - 322 q^{85} + 1217 q^{89} + 1352 q^{91} + 672 q^{95} - 263 q^{97}+O(q^{100})$$ q + 7 * q^5 + 26 * q^7 - 11 * q^11 + 52 * q^13 - 46 * q^17 + 96 * q^19 + 27 * q^23 - 76 * q^25 - 16 * q^29 + 293 * q^31 + 182 * q^35 - 29 * q^37 + 472 * q^41 + 110 * q^43 - 224 * q^47 + 333 * q^49 - 754 * q^53 - 77 * q^55 + 825 * q^59 - 548 * q^61 + 364 * q^65 + 123 * q^67 + 1001 * q^71 - 1020 * q^73 - 286 * q^77 - 526 * q^79 - 158 * q^83 - 322 * q^85 + 1217 * q^89 + 1352 * q^91 + 672 * q^95 - 263 * 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 7.00000 0 26.0000 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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$11$$ $$+1$$

## 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 1584.4.a.p 1
3.b odd 2 1 176.4.a.e 1
4.b odd 2 1 396.4.a.e 1
12.b even 2 1 44.4.a.a 1
24.f even 2 1 704.4.a.j 1
24.h odd 2 1 704.4.a.c 1
33.d even 2 1 1936.4.a.m 1
60.h even 2 1 1100.4.a.d 1
60.l odd 4 2 1100.4.b.c 2
84.h odd 2 1 2156.4.a.b 1
132.d odd 2 1 484.4.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
44.4.a.a 1 12.b even 2 1
176.4.a.e 1 3.b odd 2 1
396.4.a.e 1 4.b odd 2 1
484.4.a.a 1 132.d odd 2 1
704.4.a.c 1 24.h odd 2 1
704.4.a.j 1 24.f even 2 1
1100.4.a.d 1 60.h even 2 1
1100.4.b.c 2 60.l odd 4 2
1584.4.a.p 1 1.a even 1 1 trivial
1936.4.a.m 1 33.d even 2 1
2156.4.a.b 1 84.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1584))$$:

 $$T_{5} - 7$$ T5 - 7 $$T_{7} - 26$$ T7 - 26

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T - 7$$
$7$ $$T - 26$$
$11$ $$T + 11$$
$13$ $$T - 52$$
$17$ $$T + 46$$
$19$ $$T - 96$$
$23$ $$T - 27$$
$29$ $$T + 16$$
$31$ $$T - 293$$
$37$ $$T + 29$$
$41$ $$T - 472$$
$43$ $$T - 110$$
$47$ $$T + 224$$
$53$ $$T + 754$$
$59$ $$T - 825$$
$61$ $$T + 548$$
$67$ $$T - 123$$
$71$ $$T - 1001$$
$73$ $$T + 1020$$
$79$ $$T + 526$$
$83$ $$T + 158$$
$89$ $$T - 1217$$
$97$ $$T + 263$$