# Properties

 Label 9.4.a.a Level $9$ Weight $4$ Character orbit 9.a Self dual yes Analytic conductor $0.531$ Analytic rank $0$ Dimension $1$ CM discriminant -3 Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9,4,Mod(1,9)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("9.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$9 = 3^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 9.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: $$0.531017190052$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$+1$$ Sato-Tate group: $N(\mathrm{U}(1))$

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q - 8 q^{4} + 20 q^{7}+O(q^{10})$$ q - 8 * q^4 + 20 * q^7 $$q - 8 q^{4} + 20 q^{7} - 70 q^{13} + 64 q^{16} + 56 q^{19} - 125 q^{25} - 160 q^{28} + 308 q^{31} + 110 q^{37} - 520 q^{43} + 57 q^{49} + 560 q^{52} + 182 q^{61} - 512 q^{64} - 880 q^{67} + 1190 q^{73} - 448 q^{76} + 884 q^{79} - 1400 q^{91} - 1330 q^{97}+O(q^{100})$$ q - 8 * q^4 + 20 * q^7 - 70 * q^13 + 64 * q^16 + 56 * q^19 - 125 * q^25 - 160 * q^28 + 308 * q^31 + 110 * q^37 - 520 * q^43 + 57 * q^49 + 560 * q^52 + 182 * q^61 - 512 * q^64 - 880 * q^67 + 1190 * q^73 - 448 * q^76 + 884 * q^79 - 1400 * q^91 - 1330 * q^97

## Expression as an eta quotient

$$f(z) = \eta(3z)^{8}=q\prod_{n=1}^\infty(1 - q^{3n})^{8}$$

## 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 −8.00000 0 0 20.0000 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
$$3$$ $$+1$$

## 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 9.4.a.a 1
3.b odd 2 1 CM 9.4.a.a 1
4.b odd 2 1 144.4.a.d 1
5.b even 2 1 225.4.a.d 1
5.c odd 4 2 225.4.b.g 2
7.b odd 2 1 441.4.a.f 1
7.c even 3 2 441.4.e.i 2
7.d odd 6 2 441.4.e.j 2
8.b even 2 1 576.4.a.m 1
8.d odd 2 1 576.4.a.l 1
9.c even 3 2 81.4.c.b 2
9.d odd 6 2 81.4.c.b 2
11.b odd 2 1 1089.4.a.g 1
12.b even 2 1 144.4.a.d 1
13.b even 2 1 1521.4.a.g 1
15.d odd 2 1 225.4.a.d 1
15.e even 4 2 225.4.b.g 2
21.c even 2 1 441.4.a.f 1
21.g even 6 2 441.4.e.j 2
21.h odd 6 2 441.4.e.i 2
24.f even 2 1 576.4.a.l 1
24.h odd 2 1 576.4.a.m 1
33.d even 2 1 1089.4.a.g 1
39.d odd 2 1 1521.4.a.g 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.4.a.a 1 1.a even 1 1 trivial
9.4.a.a 1 3.b odd 2 1 CM
81.4.c.b 2 9.c even 3 2
81.4.c.b 2 9.d odd 6 2
144.4.a.d 1 4.b odd 2 1
144.4.a.d 1 12.b even 2 1
225.4.a.d 1 5.b even 2 1
225.4.a.d 1 15.d odd 2 1
225.4.b.g 2 5.c odd 4 2
225.4.b.g 2 15.e even 4 2
441.4.a.f 1 7.b odd 2 1
441.4.a.f 1 21.c even 2 1
441.4.e.i 2 7.c even 3 2
441.4.e.i 2 21.h odd 6 2
441.4.e.j 2 7.d odd 6 2
441.4.e.j 2 21.g even 6 2
576.4.a.l 1 8.d odd 2 1
576.4.a.l 1 24.f even 2 1
576.4.a.m 1 8.b even 2 1
576.4.a.m 1 24.h odd 2 1
1089.4.a.g 1 11.b odd 2 1
1089.4.a.g 1 33.d even 2 1
1521.4.a.g 1 13.b even 2 1
1521.4.a.g 1 39.d odd 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{4}^{\mathrm{new}}(\Gamma_0(9))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T - 20$$
$11$ $$T$$
$13$ $$T + 70$$
$17$ $$T$$
$19$ $$T - 56$$
$23$ $$T$$
$29$ $$T$$
$31$ $$T - 308$$
$37$ $$T - 110$$
$41$ $$T$$
$43$ $$T + 520$$
$47$ $$T$$
$53$ $$T$$
$59$ $$T$$
$61$ $$T - 182$$
$67$ $$T + 880$$
$71$ $$T$$
$73$ $$T - 1190$$
$79$ $$T - 884$$
$83$ $$T$$
$89$ $$T$$
$97$ $$T + 1330$$