# Properties

 Label 1728.1.bn.a Level $1728$ Weight $1$ Character orbit 1728.bn Analytic conductor $0.862$ Analytic rank $0$ Dimension $12$ Projective image $D_{18}$ CM discriminant -8 Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1728,1,Mod(353,1728)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1728, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([0, 9, 1]))

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

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

chi := DirichletCharacter("1728.353");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1728 = 2^{6} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1728.bn (of order $$18$$, degree $$6$$, 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: $$12$$ Relative dimension: $$2$$ over $$\Q(\zeta_{18})$$ Coefficient field: $$\Q(\zeta_{36})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{12} - x^{6} + 1$$ x^12 - x^6 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{18}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{18} - \cdots)$$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \zeta_{36}^{11} q^{3} - \zeta_{36}^{4} q^{9} +O(q^{10})$$ q + z^11 * q^3 - z^4 * q^9 $$q + \zeta_{36}^{11} q^{3} - \zeta_{36}^{4} q^{9} + ( - \zeta_{36}^{3} - \zeta_{36}) q^{11} + (\zeta_{36}^{16} - \zeta_{36}^{8}) q^{17} + (\zeta_{36}^{17} + \zeta_{36}^{13}) q^{19} + \zeta_{36}^{14} q^{25} - \zeta_{36}^{15} q^{27} + ( - \zeta_{36}^{14} - \zeta_{36}^{12}) q^{33} + (\zeta_{36}^{6} - \zeta_{36}^{2}) q^{41} + (\zeta_{36}^{15} + \zeta_{36}^{7}) q^{43} - \zeta_{36}^{16} q^{49} + ( - \zeta_{36}^{9} + \zeta_{36}) q^{51} + ( - \zeta_{36}^{10} - \zeta_{36}^{6}) q^{57} + ( - \zeta_{36}^{9} + \zeta_{36}^{5}) q^{59} + ( - \zeta_{36}^{5} + \zeta_{36}^{3}) q^{67} + ( - \zeta_{36}^{10} - \zeta_{36}^{2}) q^{73} - \zeta_{36}^{7} q^{75} + \zeta_{36}^{8} q^{81} + ( - \zeta_{36}^{17} - \zeta_{36}^{11}) q^{83} + (\zeta_{36}^{12} - 1) q^{89} + ( - \zeta_{36}^{12} + \zeta_{36}^{10}) q^{97} + (\zeta_{36}^{7} + \zeta_{36}^{5}) q^{99} +O(q^{100})$$ q + z^11 * q^3 - z^4 * q^9 + (-z^3 - z) * q^11 + (z^16 - z^8) * q^17 + (z^17 + z^13) * q^19 + z^14 * q^25 - z^15 * q^27 + (-z^14 - z^12) * q^33 + (z^6 - z^2) * q^41 + (z^15 + z^7) * q^43 - z^16 * q^49 + (-z^9 + z) * q^51 + (-z^10 - z^6) * q^57 + (-z^9 + z^5) * q^59 + (-z^5 + z^3) * q^67 + (-z^10 - z^2) * q^73 - z^7 * q^75 + z^8 * q^81 + (-z^17 - z^11) * q^83 + (z^12 - 1) * q^89 + (-z^12 + z^10) * q^97 + (z^7 + z^5) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q+O(q^{10})$$ 12 * q $$12 q + 6 q^{33} + 6 q^{41} - 6 q^{57} - 18 q^{89} + 6 q^{97}+O(q^{100})$$ 12 * q + 6 * q^33 + 6 * q^41 - 6 * q^57 - 18 * q^89 + 6 * q^97

## Character values

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

 $$n$$ $$325$$ $$703$$ $$1217$$ $$\chi(n)$$ $$-1$$ $$1$$ $$\zeta_{36}^{10}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
353.1
 −0.342020 + 0.939693i 0.342020 − 0.939693i 0.984808 + 0.173648i −0.984808 − 0.173648i 0.984808 − 0.173648i −0.984808 + 0.173648i −0.342020 − 0.939693i 0.342020 + 0.939693i 0.642788 + 0.766044i −0.642788 − 0.766044i 0.642788 − 0.766044i −0.642788 + 0.766044i
0 −0.642788 + 0.766044i 0 0 0 0 0 −0.173648 0.984808i 0
353.2 0 0.642788 0.766044i 0 0 0 0 0 −0.173648 0.984808i 0
545.1 0 −0.342020 + 0.939693i 0 0 0 0 0 −0.766044 0.642788i 0
545.2 0 0.342020 0.939693i 0 0 0 0 0 −0.766044 0.642788i 0
929.1 0 −0.342020 0.939693i 0 0 0 0 0 −0.766044 + 0.642788i 0
929.2 0 0.342020 + 0.939693i 0 0 0 0 0 −0.766044 + 0.642788i 0
1121.1 0 −0.642788 0.766044i 0 0 0 0 0 −0.173648 + 0.984808i 0
1121.2 0 0.642788 + 0.766044i 0 0 0 0 0 −0.173648 + 0.984808i 0
1505.1 0 −0.984808 0.173648i 0 0 0 0 0 0.939693 + 0.342020i 0
1505.2 0 0.984808 + 0.173648i 0 0 0 0 0 0.939693 + 0.342020i 0
1697.1 0 −0.984808 + 0.173648i 0 0 0 0 0 0.939693 0.342020i 0
1697.2 0 0.984808 0.173648i 0 0 0 0 0 0.939693 0.342020i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 353.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by $$\Q(\sqrt{-2})$$
4.b odd 2 1 inner
8.b even 2 1 inner
27.f odd 18 1 inner
108.l even 18 1 inner
216.v even 18 1 inner
216.x odd 18 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1728.1.bn.a 12
4.b odd 2 1 inner 1728.1.bn.a 12
8.b even 2 1 inner 1728.1.bn.a 12
8.d odd 2 1 CM 1728.1.bn.a 12
27.f odd 18 1 inner 1728.1.bn.a 12
108.l even 18 1 inner 1728.1.bn.a 12
216.v even 18 1 inner 1728.1.bn.a 12
216.x odd 18 1 inner 1728.1.bn.a 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1728.1.bn.a 12 1.a even 1 1 trivial
1728.1.bn.a 12 4.b odd 2 1 inner
1728.1.bn.a 12 8.b even 2 1 inner
1728.1.bn.a 12 8.d odd 2 1 CM
1728.1.bn.a 12 27.f odd 18 1 inner
1728.1.bn.a 12 108.l even 18 1 inner
1728.1.bn.a 12 216.v even 18 1 inner
1728.1.bn.a 12 216.x odd 18 1 inner

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12}$$
$3$ $$T^{12} - T^{6} + 1$$
$5$ $$T^{12}$$
$7$ $$T^{12}$$
$11$ $$T^{12} - 3 T^{10} + 30 T^{6} + 36 T^{4} + \cdots + 9$$
$13$ $$T^{12}$$
$17$ $$(T^{6} - 3 T^{4} + 9 T^{2} + 9 T + 3)^{2}$$
$19$ $$T^{12} - 6 T^{10} + 27 T^{8} - 52 T^{6} + \cdots + 1$$
$23$ $$T^{12}$$
$29$ $$T^{12}$$
$31$ $$T^{12}$$
$37$ $$T^{12}$$
$41$ $$(T^{6} - 3 T^{5} + 6 T^{4} - 6 T^{3} + 3)^{2}$$
$43$ $$T^{12} - 3 T^{10} - 6 T^{8} + 8 T^{6} + \cdots + 1$$
$47$ $$T^{12}$$
$53$ $$T^{12}$$
$59$ $$T^{12} + 6 T^{10} + 9 T^{8} + 3 T^{6} + \cdots + 9$$
$61$ $$T^{12}$$
$67$ $$T^{12} - 3 T^{10} - 6 T^{8} + 8 T^{6} + \cdots + 1$$
$71$ $$T^{12}$$
$73$ $$(T^{6} + 3 T^{4} + 2 T^{3} + 9 T^{2} + 3 T + 1)^{2}$$
$79$ $$T^{12}$$
$83$ $$T^{12} + 27T^{6} + 729$$
$89$ $$(T^{2} + 3 T + 3)^{6}$$
$97$ $$(T^{6} - 3 T^{5} + 6 T^{4} - 8 T^{3} + 12 T^{2} + \cdots + 1)^{2}$$