# Properties

 Label 1323.4.a.f Level $1323$ Weight $4$ Character orbit 1323.a Self dual yes Analytic conductor $78.060$ 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] = [1323,4,Mod(1,1323)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1323, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1323.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1323 = 3^{3} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1323.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: $$78.0595269376$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 189) 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)

 $$f(q)$$ $$=$$ $$q - 8 q^{4}+O(q^{10})$$ q - 8 * q^4 $$q - 8 q^{4} - 89 q^{13} + 64 q^{16} - 56 q^{19} - 125 q^{25} + 289 q^{31} - 433 q^{37} + 71 q^{43} + 712 q^{52} - 719 q^{61} - 512 q^{64} + 1007 q^{67} - 1190 q^{73} + 448 q^{76} + 503 q^{79} + 523 q^{97}+O(q^{100})$$ q - 8 * q^4 - 89 * q^13 + 64 * q^16 - 56 * q^19 - 125 * q^25 + 289 * q^31 - 433 * q^37 + 71 * q^43 + 712 * q^52 - 719 * q^61 - 512 * q^64 + 1007 * q^67 - 1190 * q^73 + 448 * q^76 + 503 * q^79 + 523 * 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
 0
0 0 −8.00000 0 0 0 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
$$3$$ $$-1$$
$$7$$ $$-1$$

## 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 1323.4.a.f 1
3.b odd 2 1 CM 1323.4.a.f 1
7.b odd 2 1 1323.4.a.i 1
7.d odd 6 2 189.4.e.c 2
21.c even 2 1 1323.4.a.i 1
21.g even 6 2 189.4.e.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.4.e.c 2 7.d odd 6 2
189.4.e.c 2 21.g even 6 2
1323.4.a.f 1 1.a even 1 1 trivial
1323.4.a.f 1 3.b odd 2 1 CM
1323.4.a.i 1 7.b odd 2 1
1323.4.a.i 1 21.c even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1323))$$:

 $$T_{2}$$ T2 $$T_{5}$$ T5 $$T_{13} + 89$$ T13 + 89

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T$$
$11$ $$T$$
$13$ $$T + 89$$
$17$ $$T$$
$19$ $$T + 56$$
$23$ $$T$$
$29$ $$T$$
$31$ $$T - 289$$
$37$ $$T + 433$$
$41$ $$T$$
$43$ $$T - 71$$
$47$ $$T$$
$53$ $$T$$
$59$ $$T$$
$61$ $$T + 719$$
$67$ $$T - 1007$$
$71$ $$T$$
$73$ $$T + 1190$$
$79$ $$T - 503$$
$83$ $$T$$
$89$ $$T$$
$97$ $$T - 523$$