# Properties

 Label 1728.2.a.m Level $1728$ Weight $2$ Character orbit 1728.a Self dual yes Analytic conductor $13.798$ 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] = [1728,2,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, 2, names="a")

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

chi := DirichletCharacter("1728.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1728 = 2^{6} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$2$$ 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: $$13.7981494693$$ 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)

 $$f(q)$$ $$=$$ $$q - 5 q^{7}+O(q^{10})$$ q - 5 * q^7 $$q - 5 q^{7} + 7 q^{13} - q^{19} - 5 q^{25} + 4 q^{31} + q^{37} + 8 q^{43} + 18 q^{49} + 13 q^{61} + 11 q^{67} + 17 q^{73} + 13 q^{79} - 35 q^{91} + 5 q^{97}+O(q^{100})$$ q - 5 * q^7 + 7 * q^13 - q^19 - 5 * q^25 + 4 * q^31 + q^37 + 8 * q^43 + 18 * q^49 + 13 * q^61 + 11 * q^67 + 17 * q^73 + 13 * q^79 - 35 * q^91 + 5 * 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 −5.00000 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

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 1728.2.a.m 1
3.b odd 2 1 CM 1728.2.a.m 1
4.b odd 2 1 1728.2.a.p 1
8.b even 2 1 432.2.a.d 1
8.d odd 2 1 108.2.a.a 1
12.b even 2 1 1728.2.a.p 1
24.f even 2 1 108.2.a.a 1
24.h odd 2 1 432.2.a.d 1
40.e odd 2 1 2700.2.a.b 1
40.k even 4 2 2700.2.d.g 2
56.e even 2 1 5292.2.a.j 1
72.j odd 6 2 1296.2.i.j 2
72.l even 6 2 324.2.e.b 2
72.n even 6 2 1296.2.i.j 2
72.p odd 6 2 324.2.e.b 2
120.m even 2 1 2700.2.a.b 1
120.q odd 4 2 2700.2.d.g 2
168.e odd 2 1 5292.2.a.j 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.2.a.a 1 8.d odd 2 1
108.2.a.a 1 24.f even 2 1
324.2.e.b 2 72.l even 6 2
324.2.e.b 2 72.p odd 6 2
432.2.a.d 1 8.b even 2 1
432.2.a.d 1 24.h odd 2 1
1296.2.i.j 2 72.j odd 6 2
1296.2.i.j 2 72.n even 6 2
1728.2.a.m 1 1.a even 1 1 trivial
1728.2.a.m 1 3.b odd 2 1 CM
1728.2.a.p 1 4.b odd 2 1
1728.2.a.p 1 12.b even 2 1
2700.2.a.b 1 40.e odd 2 1
2700.2.a.b 1 120.m even 2 1
2700.2.d.g 2 40.k even 4 2
2700.2.d.g 2 120.q odd 4 2
5292.2.a.j 1 56.e even 2 1
5292.2.a.j 1 168.e odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1728))$$:

 $$T_{5}$$ T5 $$T_{7} + 5$$ T7 + 5 $$T_{11}$$ T11

## Hecke characteristic polynomials

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