# Properties

 Label 1323.4.a.j Level $1323$ Weight $4$ Character orbit 1323.a Self dual yes Analytic conductor $78.060$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# 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: $$1$$ 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: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q - 8 q^{4} + 21 q^{5}+O(q^{10})$$ q - 8 * q^4 + 21 * q^5 $$q - 8 q^{4} + 21 q^{5} + 21 q^{11} - 2 q^{13} + 64 q^{16} - 42 q^{17} - 119 q^{19} - 168 q^{20} - 147 q^{23} + 316 q^{25} - 210 q^{29} - 65 q^{31} - 97 q^{37} - 399 q^{41} + 92 q^{43} - 168 q^{44} + 252 q^{47} + 16 q^{52} + 672 q^{53} + 441 q^{55} - 504 q^{59} - 632 q^{61} - 512 q^{64} - 42 q^{65} + 650 q^{67} + 336 q^{68} - 567 q^{71} + 448 q^{73} + 952 q^{76} - 484 q^{79} + 1344 q^{80} + 462 q^{83} - 882 q^{85} - 1407 q^{89} + 1176 q^{92} - 2499 q^{95} - 488 q^{97}+O(q^{100})$$ q - 8 * q^4 + 21 * q^5 + 21 * q^11 - 2 * q^13 + 64 * q^16 - 42 * q^17 - 119 * q^19 - 168 * q^20 - 147 * q^23 + 316 * q^25 - 210 * q^29 - 65 * q^31 - 97 * q^37 - 399 * q^41 + 92 * q^43 - 168 * q^44 + 252 * q^47 + 16 * q^52 + 672 * q^53 + 441 * q^55 - 504 * q^59 - 632 * q^61 - 512 * q^64 - 42 * q^65 + 650 * q^67 + 336 * q^68 - 567 * q^71 + 448 * q^73 + 952 * q^76 - 484 * q^79 + 1344 * q^80 + 462 * q^83 - 882 * q^85 - 1407 * q^89 + 1176 * q^92 - 2499 * q^95 - 488 * 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 21.0000 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

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1323.4.a.j 1
3.b odd 2 1 1323.4.a.e 1
7.b odd 2 1 189.4.a.b 1
21.c even 2 1 189.4.a.c yes 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.4.a.b 1 7.b odd 2 1
189.4.a.c yes 1 21.c even 2 1
1323.4.a.e 1 3.b odd 2 1
1323.4.a.j 1 1.a even 1 1 trivial

## 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 $$T_{5} - 21$$ T5 - 21 $$T_{13} + 2$$ T13 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T - 21$$
$7$ $$T$$
$11$ $$T - 21$$
$13$ $$T + 2$$
$17$ $$T + 42$$
$19$ $$T + 119$$
$23$ $$T + 147$$
$29$ $$T + 210$$
$31$ $$T + 65$$
$37$ $$T + 97$$
$41$ $$T + 399$$
$43$ $$T - 92$$
$47$ $$T - 252$$
$53$ $$T - 672$$
$59$ $$T + 504$$
$61$ $$T + 632$$
$67$ $$T - 650$$
$71$ $$T + 567$$
$73$ $$T - 448$$
$79$ $$T + 484$$
$83$ $$T - 462$$
$89$ $$T + 1407$$
$97$ $$T + 488$$