# Properties

 Label 1728.4.a.q Level $1728$ Weight $4$ Character orbit 1728.a Self dual yes Analytic conductor $101.955$ Analytic rank $0$ Dimension $1$ CM discriminant -3 Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1728,4,Mod(1,1728)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1728, 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("1728.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1728 = 2^{6} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1728.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: $$101.955300490$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 108) 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)

 $$f(q)$$ $$=$$ $$q + 17 q^{7}+O(q^{10})$$ q + 17 * q^7 $$q + 17 q^{7} - 89 q^{13} - 107 q^{19} - 125 q^{25} + 308 q^{31} + 433 q^{37} + 520 q^{43} - 54 q^{49} + 901 q^{61} - 1007 q^{67} - 271 q^{73} + 503 q^{79} - 1513 q^{91} + 1853 q^{97}+O(q^{100})$$ q + 17 * q^7 - 89 * q^13 - 107 * q^19 - 125 * q^25 + 308 * q^31 + 433 * q^37 + 520 * q^43 - 54 * q^49 + 901 * q^61 - 1007 * q^67 - 271 * q^73 + 503 * q^79 - 1513 * q^91 + 1853 * 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 17.0000 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
$$2$$ $$+1$$
$$3$$ $$+1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1728.4.a.q 1
3.b odd 2 1 CM 1728.4.a.q 1
4.b odd 2 1 1728.4.a.p 1
8.b even 2 1 108.4.a.c 1
8.d odd 2 1 432.4.a.g 1
12.b even 2 1 1728.4.a.p 1
24.f even 2 1 432.4.a.g 1
24.h odd 2 1 108.4.a.c 1
72.j odd 6 2 324.4.e.d 2
72.n even 6 2 324.4.e.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.4.a.c 1 8.b even 2 1
108.4.a.c 1 24.h odd 2 1
324.4.e.d 2 72.j odd 6 2
324.4.e.d 2 72.n even 6 2
432.4.a.g 1 8.d odd 2 1
432.4.a.g 1 24.f even 2 1
1728.4.a.p 1 4.b odd 2 1
1728.4.a.p 1 12.b even 2 1
1728.4.a.q 1 1.a even 1 1 trivial
1728.4.a.q 1 3.b odd 2 1 CM

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1728))$$:

 $$T_{5}$$ T5 $$T_{7} - 17$$ T7 - 17 $$T_{11}$$ T11

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T - 17$$
$11$ $$T$$
$13$ $$T + 89$$
$17$ $$T$$
$19$ $$T + 107$$
$23$ $$T$$
$29$ $$T$$
$31$ $$T - 308$$
$37$ $$T - 433$$
$41$ $$T$$
$43$ $$T - 520$$
$47$ $$T$$
$53$ $$T$$
$59$ $$T$$
$61$ $$T - 901$$
$67$ $$T + 1007$$
$71$ $$T$$
$73$ $$T + 271$$
$79$ $$T - 503$$
$83$ $$T$$
$89$ $$T$$
$97$ $$T - 1853$$