# Properties

 Label 800.4.a.e Level $800$ Weight $4$ Character orbit 800.a Self dual yes Analytic conductor $47.202$ Analytic rank $0$ Dimension $1$ CM discriminant -4 Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [800,4,Mod(1,800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(800, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("800.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$800 = 2^{5} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 800.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: $$47.2015280046$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 160) Fricke sign: $$+1$$ Sato-Tate group: $N(\mathrm{U}(1))$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q - 27 q^{9}+O(q^{10})$$ q - 27 * q^9 $$q - 27 q^{9} - 92 q^{13} + 104 q^{17} + 130 q^{29} + 396 q^{37} + 230 q^{41} - 343 q^{49} + 572 q^{53} - 830 q^{61} + 592 q^{73} + 729 q^{81} + 1670 q^{89} + 1816 q^{97}+O(q^{100})$$ q - 27 * q^9 - 92 * q^13 + 104 * q^17 + 130 * q^29 + 396 * q^37 + 230 * q^41 - 343 * q^49 + 572 * q^53 - 830 * q^61 + 592 * q^73 + 729 * q^81 + 1670 * q^89 + 1816 * 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 0 0 0 0 −27.0000 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
$$2$$ $$-1$$
$$5$$ $$-1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 800.4.a.e 1
4.b odd 2 1 CM 800.4.a.e 1
5.b even 2 1 800.4.a.g 1
5.c odd 4 2 160.4.c.a 2
8.b even 2 1 1600.4.a.bb 1
8.d odd 2 1 1600.4.a.bb 1
15.e even 4 2 1440.4.f.e 2
20.d odd 2 1 800.4.a.g 1
20.e even 4 2 160.4.c.a 2
40.e odd 2 1 1600.4.a.z 1
40.f even 2 1 1600.4.a.z 1
40.i odd 4 2 320.4.c.e 2
40.k even 4 2 320.4.c.e 2
60.l odd 4 2 1440.4.f.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.4.c.a 2 5.c odd 4 2
160.4.c.a 2 20.e even 4 2
320.4.c.e 2 40.i odd 4 2
320.4.c.e 2 40.k even 4 2
800.4.a.e 1 1.a even 1 1 trivial
800.4.a.e 1 4.b odd 2 1 CM
800.4.a.g 1 5.b even 2 1
800.4.a.g 1 20.d odd 2 1
1440.4.f.e 2 15.e even 4 2
1440.4.f.e 2 60.l odd 4 2
1600.4.a.z 1 40.e odd 2 1
1600.4.a.z 1 40.f even 2 1
1600.4.a.bb 1 8.b even 2 1
1600.4.a.bb 1 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(800))$$:

 $$T_{3}$$ T3 $$T_{11}$$ T11 $$T_{13} + 92$$ T13 + 92

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T$$
$11$ $$T$$
$13$ $$T + 92$$
$17$ $$T - 104$$
$19$ $$T$$
$23$ $$T$$
$29$ $$T - 130$$
$31$ $$T$$
$37$ $$T - 396$$
$41$ $$T - 230$$
$43$ $$T$$
$47$ $$T$$
$53$ $$T - 572$$
$59$ $$T$$
$61$ $$T + 830$$
$67$ $$T$$
$71$ $$T$$
$73$ $$T - 592$$
$79$ $$T$$
$83$ $$T$$
$89$ $$T - 1670$$
$97$ $$T - 1816$$