# Properties

 Label 256.4.a.a Level $256$ Weight $4$ Character orbit 256.a Self dual yes Analytic conductor $15.104$ Analytic rank $1$ Dimension $1$ CM discriminant -8 Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [256,4,Mod(1,256)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(256, 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("256.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$256 = 2^{8}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 256.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: $$15.1044889615$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 64) 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)

 $$f(q)$$ $$=$$ $$q - 10 q^{3} + 73 q^{9}+O(q^{10})$$ q - 10 * q^3 + 73 * q^9 $$q - 10 q^{3} + 73 q^{9} + 18 q^{11} + 90 q^{17} - 106 q^{19} - 125 q^{25} - 460 q^{27} - 180 q^{33} - 522 q^{41} + 290 q^{43} - 343 q^{49} - 900 q^{51} + 1060 q^{57} - 846 q^{59} + 70 q^{67} + 430 q^{73} + 1250 q^{75} + 2629 q^{81} + 1350 q^{83} - 1026 q^{89} - 1910 q^{97} + 1314 q^{99}+O(q^{100})$$ q - 10 * q^3 + 73 * q^9 + 18 * q^11 + 90 * q^17 - 106 * q^19 - 125 * q^25 - 460 * q^27 - 180 * q^33 - 522 * q^41 + 290 * q^43 - 343 * q^49 - 900 * q^51 + 1060 * q^57 - 846 * q^59 + 70 * q^67 + 430 * q^73 + 1250 * q^75 + 2629 * q^81 + 1350 * q^83 - 1026 * q^89 - 1910 * q^97 + 1314 * q^99

## 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 −10.0000 0 0 0 0 0 73.0000 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
8.d odd 2 1 CM by $$\Q(\sqrt{-2})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 256.4.a.a 1
3.b odd 2 1 2304.4.a.h 1
4.b odd 2 1 256.4.a.h 1
8.b even 2 1 256.4.a.h 1
8.d odd 2 1 CM 256.4.a.a 1
12.b even 2 1 2304.4.a.i 1
16.e even 4 2 64.4.b.a 2
16.f odd 4 2 64.4.b.a 2
24.f even 2 1 2304.4.a.h 1
24.h odd 2 1 2304.4.a.i 1
48.i odd 4 2 576.4.d.a 2
48.k even 4 2 576.4.d.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
64.4.b.a 2 16.e even 4 2
64.4.b.a 2 16.f odd 4 2
256.4.a.a 1 1.a even 1 1 trivial
256.4.a.a 1 8.d odd 2 1 CM
256.4.a.h 1 4.b odd 2 1
256.4.a.h 1 8.b even 2 1
576.4.d.a 2 48.i odd 4 2
576.4.d.a 2 48.k even 4 2
2304.4.a.h 1 3.b odd 2 1
2304.4.a.h 1 24.f even 2 1
2304.4.a.i 1 12.b even 2 1
2304.4.a.i 1 24.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(256))$$:

 $$T_{3} + 10$$ T3 + 10 $$T_{5}$$ T5 $$T_{7}$$ T7

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 10$$
$5$ $$T$$
$7$ $$T$$
$11$ $$T - 18$$
$13$ $$T$$
$17$ $$T - 90$$
$19$ $$T + 106$$
$23$ $$T$$
$29$ $$T$$
$31$ $$T$$
$37$ $$T$$
$41$ $$T + 522$$
$43$ $$T - 290$$
$47$ $$T$$
$53$ $$T$$
$59$ $$T + 846$$
$61$ $$T$$
$67$ $$T - 70$$
$71$ $$T$$
$73$ $$T - 430$$
$79$ $$T$$
$83$ $$T - 1350$$
$89$ $$T + 1026$$
$97$ $$T + 1910$$