# Properties

 Label 64.8.a.d Level $64$ Weight $8$ Character orbit 64.a Self dual yes Analytic conductor $19.993$ Analytic rank $1$ Dimension $1$ CM discriminant -4 Inner twists $2$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [64,8,Mod(1,64)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(64, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("64.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$64 = 2^{6}$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 64.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: $$19.9926416310$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 32) 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)

 $$f(q)$$ $$=$$ $$q + 58 q^{5} - 2187 q^{9}+O(q^{10})$$ q + 58 * q^5 - 2187 * q^9 $$q + 58 q^{5} - 2187 q^{9} + 8898 q^{13} - 40094 q^{17} - 74761 q^{25} - 233230 q^{29} - 563974 q^{37} + 9530 q^{41} - 126846 q^{45} - 823543 q^{49} + 798602 q^{53} + 3505330 q^{61} + 516084 q^{65} + 3917418 q^{73} + 4782969 q^{81} - 2325452 q^{85} + 9246170 q^{89} - 17567406 q^{97}+O(q^{100})$$ q + 58 * q^5 - 2187 * q^9 + 8898 * q^13 - 40094 * q^17 - 74761 * q^25 - 233230 * q^29 - 563974 * q^37 + 9530 * q^41 - 126846 * q^45 - 823543 * q^49 + 798602 * q^53 + 3505330 * q^61 + 516084 * q^65 + 3917418 * q^73 + 4782969 * q^81 - 2325452 * q^85 + 9246170 * q^89 - 17567406 * 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 58.0000 0 0 0 −2187.00 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
$$2$$ $$-1$$

## 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 64.8.a.d 1
3.b odd 2 1 576.8.a.n 1
4.b odd 2 1 CM 64.8.a.d 1
8.b even 2 1 32.8.a.a 1
8.d odd 2 1 32.8.a.a 1
12.b even 2 1 576.8.a.n 1
16.e even 4 2 256.8.b.d 2
16.f odd 4 2 256.8.b.d 2
24.f even 2 1 288.8.a.b 1
24.h odd 2 1 288.8.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
32.8.a.a 1 8.b even 2 1
32.8.a.a 1 8.d odd 2 1
64.8.a.d 1 1.a even 1 1 trivial
64.8.a.d 1 4.b odd 2 1 CM
256.8.b.d 2 16.e even 4 2
256.8.b.d 2 16.f odd 4 2
288.8.a.b 1 24.f even 2 1
288.8.a.b 1 24.h odd 2 1
576.8.a.n 1 3.b odd 2 1
576.8.a.n 1 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}$$ acting on $$S_{8}^{\mathrm{new}}(\Gamma_0(64))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T - 58$$
$7$ $$T$$
$11$ $$T$$
$13$ $$T - 8898$$
$17$ $$T + 40094$$
$19$ $$T$$
$23$ $$T$$
$29$ $$T + 233230$$
$31$ $$T$$
$37$ $$T + 563974$$
$41$ $$T - 9530$$
$43$ $$T$$
$47$ $$T$$
$53$ $$T - 798602$$
$59$ $$T$$
$61$ $$T - 3505330$$
$67$ $$T$$
$71$ $$T$$
$73$ $$T - 3917418$$
$79$ $$T$$
$83$ $$T$$
$89$ $$T - 9246170$$
$97$ $$T + 17567406$$
show more
show less