# Properties

 Label 1584.2.o.a Level $1584$ Weight $2$ Character orbit 1584.o Analytic conductor $12.648$ Analytic rank $0$ Dimension $2$ CM discriminant -11 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1584,2,Mod(703,1584)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1584, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1584.703");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1584 = 2^{4} \cdot 3^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1584.o (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: no Analytic conductor: $$12.6483036802$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-11})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 3$$ x^2 - x + 3 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 176) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{-11}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 3 q^{5}+O(q^{10})$$ q - 3 * q^5 $$q - 3 q^{5} - \beta q^{11} + \beta q^{23} + 4 q^{25} + 3 \beta q^{31} + 7 q^{37} + 2 \beta q^{47} - 7 q^{49} - 6 q^{53} + 3 \beta q^{55} + \beta q^{59} + 3 \beta q^{67} + 5 \beta q^{71} + 9 q^{89} - 17 q^{97} +O(q^{100})$$ q - 3 * q^5 - b * q^11 + b * q^23 + 4 * q^25 + 3*b * q^31 + 7 * q^37 + 2*b * q^47 - 7 * q^49 - 6 * q^53 + 3*b * q^55 + b * q^59 + 3*b * q^67 + 5*b * q^71 + 9 * q^89 - 17 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 6 q^{5}+O(q^{10})$$ 2 * q - 6 * q^5 $$2 q - 6 q^{5} + 8 q^{25} + 14 q^{37} - 14 q^{49} - 12 q^{53} + 18 q^{89} - 34 q^{97}+O(q^{100})$$ 2 * q - 6 * q^5 + 8 * q^25 + 14 * q^37 - 14 * q^49 - 12 * q^53 + 18 * q^89 - 34 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1584\mathbb{Z}\right)^\times$$.

 $$n$$ $$145$$ $$353$$ $$991$$ $$1189$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
703.1
 0.5 + 1.65831i 0.5 − 1.65831i
0 0 0 −3.00000 0 0 0 0 0
703.2 0 0 0 −3.00000 0 0 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
11.b odd 2 1 CM by $$\Q(\sqrt{-11})$$
4.b odd 2 1 inner
44.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1584.2.o.a 2
3.b odd 2 1 176.2.e.a 2
4.b odd 2 1 inner 1584.2.o.a 2
11.b odd 2 1 CM 1584.2.o.a 2
12.b even 2 1 176.2.e.a 2
24.f even 2 1 704.2.e.a 2
24.h odd 2 1 704.2.e.a 2
33.d even 2 1 176.2.e.a 2
44.c even 2 1 inner 1584.2.o.a 2
48.i odd 4 2 2816.2.g.a 4
48.k even 4 2 2816.2.g.a 4
132.d odd 2 1 176.2.e.a 2
264.m even 2 1 704.2.e.a 2
264.p odd 2 1 704.2.e.a 2
528.s odd 4 2 2816.2.g.a 4
528.x even 4 2 2816.2.g.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
176.2.e.a 2 3.b odd 2 1
176.2.e.a 2 12.b even 2 1
176.2.e.a 2 33.d even 2 1
176.2.e.a 2 132.d odd 2 1
704.2.e.a 2 24.f even 2 1
704.2.e.a 2 24.h odd 2 1
704.2.e.a 2 264.m even 2 1
704.2.e.a 2 264.p odd 2 1
1584.2.o.a 2 1.a even 1 1 trivial
1584.2.o.a 2 4.b odd 2 1 inner
1584.2.o.a 2 11.b odd 2 1 CM
1584.2.o.a 2 44.c even 2 1 inner
2816.2.g.a 4 48.i odd 4 2
2816.2.g.a 4 48.k even 4 2
2816.2.g.a 4 528.s odd 4 2
2816.2.g.a 4 528.x even 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1584, [\chi])$$:

 $$T_{5} + 3$$ T5 + 3 $$T_{83}$$ T83

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$(T + 3)^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 11$$
$13$ $$T^{2}$$
$17$ $$T^{2}$$
$19$ $$T^{2}$$
$23$ $$T^{2} + 11$$
$29$ $$T^{2}$$
$31$ $$T^{2} + 99$$
$37$ $$(T - 7)^{2}$$
$41$ $$T^{2}$$
$43$ $$T^{2}$$
$47$ $$T^{2} + 44$$
$53$ $$(T + 6)^{2}$$
$59$ $$T^{2} + 11$$
$61$ $$T^{2}$$
$67$ $$T^{2} + 99$$
$71$ $$T^{2} + 275$$
$73$ $$T^{2}$$
$79$ $$T^{2}$$
$83$ $$T^{2}$$
$89$ $$(T - 9)^{2}$$
$97$ $$(T + 17)^{2}$$