# Properties

 Label 576.4.a.m Level $576$ Weight $4$ Character orbit 576.a Self dual yes Analytic conductor $33.985$ 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] = [576,4,Mod(1,576)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(576, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("576.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$576 = 2^{6} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 576.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: $$33.9851001633$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 9) 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)

 $$f(q)$$ $$=$$ $$q + 20 q^{7}+O(q^{10})$$ q + 20 * q^7 $$q + 20 q^{7} + 70 q^{13} - 56 q^{19} - 125 q^{25} + 308 q^{31} - 110 q^{37} + 520 q^{43} + 57 q^{49} - 182 q^{61} + 880 q^{67} + 1190 q^{73} + 884 q^{79} + 1400 q^{91} - 1330 q^{97}+O(q^{100})$$ q + 20 * q^7 + 70 * q^13 - 56 * q^19 - 125 * q^25 + 308 * q^31 - 110 * q^37 + 520 * q^43 + 57 * q^49 - 182 * q^61 + 880 * q^67 + 1190 * q^73 + 884 * q^79 + 1400 * q^91 - 1330 * 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 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
$$2$$ $$+1$$
$$3$$ $$+1$$

## 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 576.4.a.m 1
3.b odd 2 1 CM 576.4.a.m 1
4.b odd 2 1 576.4.a.l 1
8.b even 2 1 9.4.a.a 1
8.d odd 2 1 144.4.a.d 1
12.b even 2 1 576.4.a.l 1
24.f even 2 1 144.4.a.d 1
24.h odd 2 1 9.4.a.a 1
40.f even 2 1 225.4.a.d 1
40.i odd 4 2 225.4.b.g 2
56.h odd 2 1 441.4.a.f 1
56.j odd 6 2 441.4.e.j 2
56.p even 6 2 441.4.e.i 2
72.j odd 6 2 81.4.c.b 2
72.n even 6 2 81.4.c.b 2
88.b odd 2 1 1089.4.a.g 1
104.e even 2 1 1521.4.a.g 1
120.i odd 2 1 225.4.a.d 1
120.w even 4 2 225.4.b.g 2
168.i even 2 1 441.4.a.f 1
168.s odd 6 2 441.4.e.i 2
168.ba even 6 2 441.4.e.j 2
264.m even 2 1 1089.4.a.g 1
312.b 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 8.b even 2 1
9.4.a.a 1 24.h odd 2 1
81.4.c.b 2 72.j odd 6 2
81.4.c.b 2 72.n even 6 2
144.4.a.d 1 8.d odd 2 1
144.4.a.d 1 24.f even 2 1
225.4.a.d 1 40.f even 2 1
225.4.a.d 1 120.i odd 2 1
225.4.b.g 2 40.i odd 4 2
225.4.b.g 2 120.w even 4 2
441.4.a.f 1 56.h odd 2 1
441.4.a.f 1 168.i even 2 1
441.4.e.i 2 56.p even 6 2
441.4.e.i 2 168.s odd 6 2
441.4.e.j 2 56.j odd 6 2
441.4.e.j 2 168.ba even 6 2
576.4.a.l 1 4.b odd 2 1
576.4.a.l 1 12.b even 2 1
576.4.a.m 1 1.a even 1 1 trivial
576.4.a.m 1 3.b odd 2 1 CM
1089.4.a.g 1 88.b odd 2 1
1089.4.a.g 1 264.m even 2 1
1521.4.a.g 1 104.e even 2 1
1521.4.a.g 1 312.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(576))$$:

 $$T_{5}$$ T5 $$T_{7} - 20$$ T7 - 20 $$T_{11}$$ T11

## Hecke characteristic polynomials

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