# Properties

 Label 1728.2.a.bd Level $1728$ Weight $2$ Character orbit 1728.a Self dual yes Analytic conductor $13.798$ Analytic rank $0$ Dimension $2$ CM no 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: $$2$$ Coefficient field: $$\Q(\sqrt{13})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 3$$ x^2 - x - 3 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ 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)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{13}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta q^{5} - \beta q^{7} +O(q^{10})$$ q - b * q^5 - b * q^7 $$q - \beta q^{5} - \beta q^{7} + q^{11} - 4 q^{13} - 2 \beta q^{19} + 6 q^{23} + 8 q^{25} + 2 \beta q^{29} + \beta q^{31} + 13 q^{35} - 10 q^{37} - 2 \beta q^{41} + 2 \beta q^{43} + 10 q^{47} + 6 q^{49} + \beta q^{53} - \beta q^{55} - 4 q^{59} + 4 \beta q^{65} + 2 \beta q^{67} - 8 q^{71} - 3 q^{73} - \beta q^{77} + 4 \beta q^{79} + 9 q^{83} + 2 \beta q^{89} + 4 \beta q^{91} + 26 q^{95} + 7 q^{97} +O(q^{100})$$ q - b * q^5 - b * q^7 + q^11 - 4 * q^13 - 2*b * q^19 + 6 * q^23 + 8 * q^25 + 2*b * q^29 + b * q^31 + 13 * q^35 - 10 * q^37 - 2*b * q^41 + 2*b * q^43 + 10 * q^47 + 6 * q^49 + b * q^53 - b * q^55 - 4 * q^59 + 4*b * q^65 + 2*b * q^67 - 8 * q^71 - 3 * q^73 - b * q^77 + 4*b * q^79 + 9 * q^83 + 2*b * q^89 + 4*b * q^91 + 26 * q^95 + 7 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q+O(q^{10})$$ 2 * q $$2 q + 2 q^{11} - 8 q^{13} + 12 q^{23} + 16 q^{25} + 26 q^{35} - 20 q^{37} + 20 q^{47} + 12 q^{49} - 8 q^{59} - 16 q^{71} - 6 q^{73} + 18 q^{83} + 52 q^{95} + 14 q^{97}+O(q^{100})$$ 2 * q + 2 * q^11 - 8 * q^13 + 12 * q^23 + 16 * q^25 + 26 * q^35 - 20 * q^37 + 20 * q^47 + 12 * q^49 - 8 * q^59 - 16 * q^71 - 6 * q^73 + 18 * q^83 + 52 * q^95 + 14 * 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
 2.30278 −1.30278
0 0 0 −3.60555 0 −3.60555 0 0 0
1.2 0 0 0 3.60555 0 3.60555 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
12.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1728.2.a.bd 2
3.b odd 2 1 1728.2.a.bc 2
4.b odd 2 1 1728.2.a.bc 2
8.b even 2 1 864.2.a.m 2
8.d odd 2 1 864.2.a.n yes 2
12.b even 2 1 inner 1728.2.a.bd 2
24.f even 2 1 864.2.a.m 2
24.h odd 2 1 864.2.a.n yes 2
72.j odd 6 2 2592.2.i.bb 4
72.l even 6 2 2592.2.i.bc 4
72.n even 6 2 2592.2.i.bc 4
72.p odd 6 2 2592.2.i.bb 4

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

## Hecke kernels

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

 $$T_{5}^{2} - 13$$ T5^2 - 13 $$T_{7}^{2} - 13$$ T7^2 - 13 $$T_{11} - 1$$ T11 - 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 13$$
$7$ $$T^{2} - 13$$
$11$ $$(T - 1)^{2}$$
$13$ $$(T + 4)^{2}$$
$17$ $$T^{2}$$
$19$ $$T^{2} - 52$$
$23$ $$(T - 6)^{2}$$
$29$ $$T^{2} - 52$$
$31$ $$T^{2} - 13$$
$37$ $$(T + 10)^{2}$$
$41$ $$T^{2} - 52$$
$43$ $$T^{2} - 52$$
$47$ $$(T - 10)^{2}$$
$53$ $$T^{2} - 13$$
$59$ $$(T + 4)^{2}$$
$61$ $$T^{2}$$
$67$ $$T^{2} - 52$$
$71$ $$(T + 8)^{2}$$
$73$ $$(T + 3)^{2}$$
$79$ $$T^{2} - 208$$
$83$ $$(T - 9)^{2}$$
$89$ $$T^{2} - 52$$
$97$ $$(T - 7)^{2}$$