# Properties

 Label 243.1.b.a Level $243$ Weight $1$ Character orbit 243.b Self dual yes Analytic conductor $0.121$ Analytic rank $0$ Dimension $1$ Projective image $D_{3}$ CM discriminant -3 Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [243,1,Mod(242,243)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(243, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("243.242");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$243 = 3^{5}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 243.b (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: yes Analytic conductor: $$0.121272798070$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.243.1 Artin image: $S_3$ Artin field: Galois closure of 3.1.243.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q + q^{4} - q^{7}+O(q^{10})$$ q + q^4 - q^7 $$q + q^{4} - q^{7} - q^{13} + q^{16} - q^{19} + q^{25} - q^{28} - q^{31} - q^{37} - q^{43} - q^{52} + 2 q^{61} + q^{64} + 2 q^{67} + 2 q^{73} - q^{76} - q^{79} + q^{91} - q^{97}+O(q^{100})$$ q + q^4 - q^7 - q^13 + q^16 - q^19 + q^25 - q^28 - q^31 - q^37 - q^43 - q^52 + 2 * q^61 + q^64 + 2 * q^67 + 2 * q^73 - q^76 - q^79 + q^91 - q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/243\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
242.1
 0
0 0 1.00000 0 0 −1.00000 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 $$\Q(\sqrt{-3})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 243.1.b.a 1
3.b odd 2 1 CM 243.1.b.a 1
4.b odd 2 1 3888.1.e.b 1
9.c even 3 2 243.1.d.a 2
9.d odd 6 2 243.1.d.a 2
12.b even 2 1 3888.1.e.b 1
27.e even 9 6 729.1.f.a 6
27.f odd 18 6 729.1.f.a 6
36.f odd 6 2 3888.1.q.b 2
36.h even 6 2 3888.1.q.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
243.1.b.a 1 1.a even 1 1 trivial
243.1.b.a 1 3.b odd 2 1 CM
243.1.d.a 2 9.c even 3 2
243.1.d.a 2 9.d odd 6 2
729.1.f.a 6 27.e even 9 6
729.1.f.a 6 27.f odd 18 6
3888.1.e.b 1 4.b odd 2 1
3888.1.e.b 1 12.b even 2 1
3888.1.q.b 2 36.f odd 6 2
3888.1.q.b 2 36.h even 6 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(243, [\chi])$$.

## Hecke characteristic polynomials

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