# Properties

 Label 256.9.d.a Level $256$ Weight $9$ Character orbit 256.d Analytic conductor $104.289$ Analytic rank $0$ Dimension $2$ CM discriminant -4 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [256,9,Mod(127,256)]

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([1, 1]))

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("256.127");

S:= CuspForms(chi, 9);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$256 = 2^{8}$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 256.d (of order $$2$$, degree $$1$$, not 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: $$104.288924176$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 4) 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 $$\beta = 2i$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 527 \beta q^{5} - 6561 q^{9} +O(q^{10})$$ q - 527*b * q^5 - 6561 * q^9 $$q - 527 \beta q^{5} - 6561 q^{9} + 239 \beta q^{13} - 63358 q^{17} - 720291 q^{25} + 703919 \beta q^{29} + 462961 \beta q^{37} - 3577922 q^{41} + 3457647 \beta q^{45} + 5764801 q^{49} - 4810319 \beta q^{53} - 10361041 \beta q^{61} + 503812 q^{65} + 54717118 q^{73} + 43046721 q^{81} + 33389666 \beta q^{85} + 30265918 q^{89} - 173379838 q^{97} +O(q^{100})$$ q - 527*b * q^5 - 6561 * q^9 + 239*b * q^13 - 63358 * q^17 - 720291 * q^25 + 703919*b * q^29 + 462961*b * q^37 - 3577922 * q^41 + 3457647*b * q^45 + 5764801 * q^49 - 4810319*b * q^53 - 10361041*b * q^61 + 503812 * q^65 + 54717118 * q^73 + 43046721 * q^81 + 33389666*b * q^85 + 30265918 * q^89 - 173379838 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 13122 q^{9}+O(q^{10})$$ 2 * q - 13122 * q^9 $$2 q - 13122 q^{9} - 126716 q^{17} - 1440582 q^{25} - 7155844 q^{41} + 11529602 q^{49} + 1007624 q^{65} + 109434236 q^{73} + 86093442 q^{81} + 60531836 q^{89} - 346759676 q^{97}+O(q^{100})$$ 2 * q - 13122 * q^9 - 126716 * q^17 - 1440582 * q^25 - 7155844 * q^41 + 11529602 * q^49 + 1007624 * q^65 + 109434236 * q^73 + 86093442 * q^81 + 60531836 * q^89 - 346759676 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/256\mathbb{Z}\right)^\times$$.

 $$n$$ $$5$$ $$255$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
127.1
 1.00000i − 1.00000i
0 0 0 1054.00i 0 0 0 −6561.00 0
127.2 0 0 0 1054.00i 0 0 0 −6561.00 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
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$
8.b even 2 1 inner
8.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 256.9.d.a 2
4.b odd 2 1 CM 256.9.d.a 2
8.b even 2 1 inner 256.9.d.a 2
8.d odd 2 1 inner 256.9.d.a 2
16.e even 4 1 4.9.b.a 1
16.e even 4 1 64.9.c.a 1
16.f odd 4 1 4.9.b.a 1
16.f odd 4 1 64.9.c.a 1
48.i odd 4 1 36.9.d.a 1
48.k even 4 1 36.9.d.a 1
80.i odd 4 1 100.9.d.a 2
80.j even 4 1 100.9.d.a 2
80.k odd 4 1 100.9.b.a 1
80.q even 4 1 100.9.b.a 1
80.s even 4 1 100.9.d.a 2
80.t odd 4 1 100.9.d.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.9.b.a 1 16.e even 4 1
4.9.b.a 1 16.f odd 4 1
36.9.d.a 1 48.i odd 4 1
36.9.d.a 1 48.k even 4 1
64.9.c.a 1 16.e even 4 1
64.9.c.a 1 16.f odd 4 1
100.9.b.a 1 80.k odd 4 1
100.9.b.a 1 80.q even 4 1
100.9.d.a 2 80.i odd 4 1
100.9.d.a 2 80.j even 4 1
100.9.d.a 2 80.s even 4 1
100.9.d.a 2 80.t odd 4 1
256.9.d.a 2 1.a even 1 1 trivial
256.9.d.a 2 4.b odd 2 1 CM
256.9.d.a 2 8.b even 2 1 inner
256.9.d.a 2 8.d odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}$$ acting on $$S_{9}^{\mathrm{new}}(256, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 1110916$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 228484$$
$17$ $$(T + 63358)^{2}$$
$19$ $$T^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2} + 1982007834244$$
$31$ $$T^{2}$$
$37$ $$T^{2} + 857331550084$$
$41$ $$(T + 3577922)^{2}$$
$43$ $$T^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2} + 92556675527044$$
$59$ $$T^{2}$$
$61$ $$T^{2} + 429404682414724$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$(T - 54717118)^{2}$$
$79$ $$T^{2}$$
$83$ $$T^{2}$$
$89$ $$(T - 30265918)^{2}$$
$97$ $$(T + 173379838)^{2}$$