# Properties

 Label 1728.1.h.a Level $1728$ Weight $1$ Character orbit 1728.h Analytic conductor $0.862$ Analytic rank $0$ Dimension $4$ Projective image $D_{6}$ CM discriminant -3 Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1728,1,Mod(161,1728)]

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, 1, 1]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("1728.161");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1728 = 2^{6} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1728.h (of order $$2$$, degree $$1$$, minimal)

## 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: no Analytic conductor: $$0.862384341830$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Projective image: $$D_{6}$$ Projective field: Galois closure of 6.0.1492992.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{12}^{5} + \zeta_{12}) q^{7}+O(q^{10})$$ q + (-z^5 + z) * q^7 $$q + ( - \zeta_{12}^{5} + \zeta_{12}) q^{7} + ( - \zeta_{12}^{4} - \zeta_{12}^{2}) q^{13} + \zeta_{12}^{3} q^{19} - q^{25} + ( - \zeta_{12}^{4} - \zeta_{12}^{2}) q^{37} + \zeta_{12}^{3} q^{43} + ( - \zeta_{12}^{4} + \zeta_{12}^{2} + 1) q^{49} + (\zeta_{12}^{4} + \zeta_{12}^{2}) q^{61} - \zeta_{12}^{3} q^{67} - q^{73} + ( - \zeta_{12}^{5} + \zeta_{12}) q^{79} + ( - \zeta_{12}^{5} - \zeta_{12}^{3} - \zeta_{12}) q^{91} + q^{97} +O(q^{100})$$ q + (-z^5 + z) * q^7 + (-z^4 - z^2) * q^13 + z^3 * q^19 - q^25 + (-z^4 - z^2) * q^37 + z^3 * q^43 + (-z^4 + z^2 + 1) * q^49 + (z^4 + z^2) * q^61 - z^3 * q^67 - q^73 + (-z^5 + z) * q^79 + (-z^5 - z^3 - z) * q^91 + q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q - 4 q^{25} + 8 q^{49} - 4 q^{73} + 4 q^{97}+O(q^{100})$$ 4 * q - 4 * q^25 + 8 * q^49 - 4 * q^73 + 4 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times$$.

 $$n$$ $$325$$ $$703$$ $$1217$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
161.1
 −0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 − 0.500000i
0 0 0 0 0 −1.73205 0 0 0
161.2 0 0 0 0 0 −1.73205 0 0 0
161.3 0 0 0 0 0 1.73205 0 0 0
161.4 0 0 0 0 0 1.73205 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
12.b even 2 1 inner
24.f even 2 1 inner
24.h odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1728.1.h.a 4
3.b odd 2 1 CM 1728.1.h.a 4
4.b odd 2 1 inner 1728.1.h.a 4
8.b even 2 1 inner 1728.1.h.a 4
8.d odd 2 1 inner 1728.1.h.a 4
12.b even 2 1 inner 1728.1.h.a 4
24.f even 2 1 inner 1728.1.h.a 4
24.h odd 2 1 inner 1728.1.h.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1728.1.h.a 4 1.a even 1 1 trivial
1728.1.h.a 4 3.b odd 2 1 CM
1728.1.h.a 4 4.b odd 2 1 inner
1728.1.h.a 4 8.b even 2 1 inner
1728.1.h.a 4 8.d odd 2 1 inner
1728.1.h.a 4 12.b even 2 1 inner
1728.1.h.a 4 24.f even 2 1 inner
1728.1.h.a 4 24.h odd 2 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(1728, [\chi])$$.

## Hecke characteristic polynomials

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