# Properties

 Label 1728.2.a.k Level $1728$ Weight $2$ Character orbit 1728.a Self dual yes Analytic conductor $13.798$ Analytic rank $1$ 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] = [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: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 864) 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)

 $$f(q)$$ $$=$$ $$q - q^{5} + 3 q^{7}+O(q^{10})$$ q - q^5 + 3 * q^7 $$q - q^{5} + 3 q^{7} - 3 q^{11} - 4 q^{17} - 6 q^{19} - 6 q^{23} - 4 q^{25} - 2 q^{29} + 9 q^{31} - 3 q^{35} + 2 q^{37} + 10 q^{41} - 6 q^{43} - 6 q^{47} + 2 q^{49} + 13 q^{53} + 3 q^{55} - 12 q^{59} - 8 q^{61} - 6 q^{67} - 12 q^{71} + 9 q^{73} - 9 q^{77} - 3 q^{83} + 4 q^{85} - 14 q^{89} + 6 q^{95} - 9 q^{97}+O(q^{100})$$ q - q^5 + 3 * q^7 - 3 * q^11 - 4 * q^17 - 6 * q^19 - 6 * q^23 - 4 * q^25 - 2 * q^29 + 9 * q^31 - 3 * q^35 + 2 * q^37 + 10 * q^41 - 6 * q^43 - 6 * q^47 + 2 * q^49 + 13 * q^53 + 3 * q^55 - 12 * q^59 - 8 * q^61 - 6 * q^67 - 12 * q^71 + 9 * q^73 - 9 * q^77 - 3 * q^83 + 4 * q^85 - 14 * q^89 + 6 * q^95 - 9 * 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 −1.00000 0 3.00000 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 1728.2.a.k 1
3.b odd 2 1 1728.2.a.t 1
4.b odd 2 1 1728.2.a.j 1
8.b even 2 1 864.2.a.h yes 1
8.d odd 2 1 864.2.a.g yes 1
12.b even 2 1 1728.2.a.q 1
24.f even 2 1 864.2.a.e 1
24.h odd 2 1 864.2.a.f yes 1
72.j odd 6 2 2592.2.i.m 2
72.l even 6 2 2592.2.i.p 2
72.n even 6 2 2592.2.i.i 2
72.p odd 6 2 2592.2.i.l 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
864.2.a.e 1 24.f even 2 1
864.2.a.f yes 1 24.h odd 2 1
864.2.a.g yes 1 8.d odd 2 1
864.2.a.h yes 1 8.b even 2 1
1728.2.a.j 1 4.b odd 2 1
1728.2.a.k 1 1.a even 1 1 trivial
1728.2.a.q 1 12.b even 2 1
1728.2.a.t 1 3.b odd 2 1
2592.2.i.i 2 72.n even 6 2
2592.2.i.l 2 72.p odd 6 2
2592.2.i.m 2 72.j odd 6 2
2592.2.i.p 2 72.l even 6 2

## Hecke kernels

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

 $$T_{5} + 1$$ T5 + 1 $$T_{7} - 3$$ T7 - 3 $$T_{11} + 3$$ T11 + 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T + 1$$
$7$ $$T - 3$$
$11$ $$T + 3$$
$13$ $$T$$
$17$ $$T + 4$$
$19$ $$T + 6$$
$23$ $$T + 6$$
$29$ $$T + 2$$
$31$ $$T - 9$$
$37$ $$T - 2$$
$41$ $$T - 10$$
$43$ $$T + 6$$
$47$ $$T + 6$$
$53$ $$T - 13$$
$59$ $$T + 12$$
$61$ $$T + 8$$
$67$ $$T + 6$$
$71$ $$T + 12$$
$73$ $$T - 9$$
$79$ $$T$$
$83$ $$T + 3$$
$89$ $$T + 14$$
$97$ $$T + 9$$