# Properties

 Label 288.2.a.a Level $288$ Weight $2$ Character orbit 288.a Self dual yes Analytic conductor $2.300$ Analytic rank $1$ Dimension $1$ CM discriminant -4 Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [288,2,Mod(1,288)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(288, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("288.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$288 = 2^{5} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 288.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: $$2.29969157821$$ Analytic rank: $$1$$ 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)

 $$f(q)$$ $$=$$ $$q - 4 q^{5}+O(q^{10})$$ q - 4 * q^5 $$q - 4 q^{5} - 6 q^{13} - 8 q^{17} + 11 q^{25} + 4 q^{29} - 2 q^{37} + 8 q^{41} - 7 q^{49} + 4 q^{53} - 10 q^{61} + 24 q^{65} + 6 q^{73} + 32 q^{85} - 16 q^{89} - 18 q^{97}+O(q^{100})$$ q - 4 * q^5 - 6 * q^13 - 8 * q^17 + 11 * q^25 + 4 * q^29 - 2 * q^37 + 8 * q^41 - 7 * q^49 + 4 * q^53 - 10 * q^61 + 24 * q^65 + 6 * q^73 + 32 * q^85 - 16 * q^89 - 18 * 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 0 −4.00000 0 0 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 288.2.a.a 1
3.b odd 2 1 288.2.a.e yes 1
4.b odd 2 1 CM 288.2.a.a 1
5.b even 2 1 7200.2.a.bf 1
5.c odd 4 2 7200.2.f.n 2
8.b even 2 1 576.2.a.i 1
8.d odd 2 1 576.2.a.i 1
9.c even 3 2 2592.2.i.x 2
9.d odd 6 2 2592.2.i.a 2
12.b even 2 1 288.2.a.e yes 1
15.d odd 2 1 7200.2.a.be 1
15.e even 4 2 7200.2.f.q 2
16.e even 4 2 2304.2.d.h 2
16.f odd 4 2 2304.2.d.h 2
20.d odd 2 1 7200.2.a.bf 1
20.e even 4 2 7200.2.f.n 2
24.f even 2 1 576.2.a.a 1
24.h odd 2 1 576.2.a.a 1
36.f odd 6 2 2592.2.i.x 2
36.h even 6 2 2592.2.i.a 2
48.i odd 4 2 2304.2.d.l 2
48.k even 4 2 2304.2.d.l 2
60.h even 2 1 7200.2.a.be 1
60.l odd 4 2 7200.2.f.q 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
288.2.a.a 1 1.a even 1 1 trivial
288.2.a.a 1 4.b odd 2 1 CM
288.2.a.e yes 1 3.b odd 2 1
288.2.a.e yes 1 12.b even 2 1
576.2.a.a 1 24.f even 2 1
576.2.a.a 1 24.h odd 2 1
576.2.a.i 1 8.b even 2 1
576.2.a.i 1 8.d odd 2 1
2304.2.d.h 2 16.e even 4 2
2304.2.d.h 2 16.f odd 4 2
2304.2.d.l 2 48.i odd 4 2
2304.2.d.l 2 48.k even 4 2
2592.2.i.a 2 9.d odd 6 2
2592.2.i.a 2 36.h even 6 2
2592.2.i.x 2 9.c even 3 2
2592.2.i.x 2 36.f odd 6 2
7200.2.a.be 1 15.d odd 2 1
7200.2.a.be 1 60.h even 2 1
7200.2.a.bf 1 5.b even 2 1
7200.2.a.bf 1 20.d odd 2 1
7200.2.f.n 2 5.c odd 4 2
7200.2.f.n 2 20.e even 4 2
7200.2.f.q 2 15.e even 4 2
7200.2.f.q 2 60.l odd 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(288))$$:

 $$T_{5} + 4$$ T5 + 4 $$T_{7}$$ T7

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T + 4$$
$7$ $$T$$
$11$ $$T$$
$13$ $$T + 6$$
$17$ $$T + 8$$
$19$ $$T$$
$23$ $$T$$
$29$ $$T - 4$$
$31$ $$T$$
$37$ $$T + 2$$
$41$ $$T - 8$$
$43$ $$T$$
$47$ $$T$$
$53$ $$T - 4$$
$59$ $$T$$
$61$ $$T + 10$$
$67$ $$T$$
$71$ $$T$$
$73$ $$T - 6$$
$79$ $$T$$
$83$ $$T$$
$89$ $$T + 16$$
$97$ $$T + 18$$