# Properties

 Label 1280.1.e.a Level $1280$ Weight $1$ Character orbit 1280.e Analytic conductor $0.639$ Analytic rank $0$ Dimension $2$ Projective image $D_{2}$ CM/RM discs -4, -20, 5 Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1280,1,Mod(639,1280)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1280, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 1]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("1280.639");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1280 = 2^{8} \cdot 5$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1280.e (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: $$0.638803216170$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 80) Projective image: $$D_{2}$$ Projective field: Galois closure of $$\Q(i, \sqrt{5})$$ Artin image: $D_4:C_2$ Artin field: Galois closure of 8.0.419430400.3

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q - i q^{5} + q^{9} +O(q^{10})$$ q - z * q^5 + q^9 $$q - i q^{5} + q^{9} - q^{25} - i q^{29} + q^{41} - i q^{45} - q^{49} + i q^{61} + q^{81} - q^{89} +O(q^{100})$$ q - z * q^5 + q^9 - q^25 - z * q^29 + q^41 - z * q^45 - q^49 + z * q^61 + q^81 - q^89 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^9 $$2 q + 2 q^{9} - 2 q^{25} + 4 q^{41} - 2 q^{49} + 2 q^{81} - 4 q^{89}+O(q^{100})$$ 2 * q + 2 * q^9 - 2 * q^25 + 4 * q^41 - 2 * q^49 + 2 * q^81 - 4 * q^89

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1280\mathbb{Z}\right)^\times$$.

 $$n$$ $$257$$ $$261$$ $$511$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
639.1
 1.00000i − 1.00000i
0 0 0 1.00000i 0 0 0 1.00000 0
639.2 0 0 0 1.00000i 0 0 0 1.00000 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 $$\Q(\sqrt{-1})$$
5.b even 2 1 RM by $$\Q(\sqrt{5})$$
20.d odd 2 1 CM by $$\Q(\sqrt{-5})$$
8.b even 2 1 inner
8.d odd 2 1 inner
40.e odd 2 1 inner
40.f even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1280.1.e.a 2
4.b odd 2 1 CM 1280.1.e.a 2
5.b even 2 1 RM 1280.1.e.a 2
8.b even 2 1 inner 1280.1.e.a 2
8.d odd 2 1 inner 1280.1.e.a 2
16.e even 4 1 80.1.h.a 1
16.e even 4 1 320.1.h.a 1
16.f odd 4 1 80.1.h.a 1
16.f odd 4 1 320.1.h.a 1
20.d odd 2 1 CM 1280.1.e.a 2
40.e odd 2 1 inner 1280.1.e.a 2
40.f even 2 1 inner 1280.1.e.a 2
48.i odd 4 1 720.1.j.a 1
48.i odd 4 1 2880.1.j.a 1
48.k even 4 1 720.1.j.a 1
48.k even 4 1 2880.1.j.a 1
80.i odd 4 1 400.1.b.a 1
80.i odd 4 1 1600.1.b.a 1
80.j even 4 1 400.1.b.a 1
80.j even 4 1 1600.1.b.a 1
80.k odd 4 1 80.1.h.a 1
80.k odd 4 1 320.1.h.a 1
80.q even 4 1 80.1.h.a 1
80.q even 4 1 320.1.h.a 1
80.s even 4 1 400.1.b.a 1
80.s even 4 1 1600.1.b.a 1
80.t odd 4 1 400.1.b.a 1
80.t odd 4 1 1600.1.b.a 1
112.j even 4 1 3920.1.j.a 1
112.l odd 4 1 3920.1.j.a 1
112.u odd 12 2 3920.1.bt.b 2
112.v even 12 2 3920.1.bt.a 2
112.w even 12 2 3920.1.bt.b 2
112.x odd 12 2 3920.1.bt.a 2
240.t even 4 1 720.1.j.a 1
240.t even 4 1 2880.1.j.a 1
240.z odd 4 1 3600.1.e.a 1
240.bb even 4 1 3600.1.e.a 1
240.bd odd 4 1 3600.1.e.a 1
240.bf even 4 1 3600.1.e.a 1
240.bm odd 4 1 720.1.j.a 1
240.bm odd 4 1 2880.1.j.a 1
560.be even 4 1 3920.1.j.a 1
560.bf odd 4 1 3920.1.j.a 1
560.cn odd 12 2 3920.1.bt.a 2
560.co even 12 2 3920.1.bt.a 2
560.cr even 12 2 3920.1.bt.b 2
560.cs odd 12 2 3920.1.bt.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.1.h.a 1 16.e even 4 1
80.1.h.a 1 16.f odd 4 1
80.1.h.a 1 80.k odd 4 1
80.1.h.a 1 80.q even 4 1
320.1.h.a 1 16.e even 4 1
320.1.h.a 1 16.f odd 4 1
320.1.h.a 1 80.k odd 4 1
320.1.h.a 1 80.q even 4 1
400.1.b.a 1 80.i odd 4 1
400.1.b.a 1 80.j even 4 1
400.1.b.a 1 80.s even 4 1
400.1.b.a 1 80.t odd 4 1
720.1.j.a 1 48.i odd 4 1
720.1.j.a 1 48.k even 4 1
720.1.j.a 1 240.t even 4 1
720.1.j.a 1 240.bm odd 4 1
1280.1.e.a 2 1.a even 1 1 trivial
1280.1.e.a 2 4.b odd 2 1 CM
1280.1.e.a 2 5.b even 2 1 RM
1280.1.e.a 2 8.b even 2 1 inner
1280.1.e.a 2 8.d odd 2 1 inner
1280.1.e.a 2 20.d odd 2 1 CM
1280.1.e.a 2 40.e odd 2 1 inner
1280.1.e.a 2 40.f even 2 1 inner
1600.1.b.a 1 80.i odd 4 1
1600.1.b.a 1 80.j even 4 1
1600.1.b.a 1 80.s even 4 1
1600.1.b.a 1 80.t odd 4 1
2880.1.j.a 1 48.i odd 4 1
2880.1.j.a 1 48.k even 4 1
2880.1.j.a 1 240.t even 4 1
2880.1.j.a 1 240.bm odd 4 1
3600.1.e.a 1 240.z odd 4 1
3600.1.e.a 1 240.bb even 4 1
3600.1.e.a 1 240.bd odd 4 1
3600.1.e.a 1 240.bf even 4 1
3920.1.j.a 1 112.j even 4 1
3920.1.j.a 1 112.l odd 4 1
3920.1.j.a 1 560.be even 4 1
3920.1.j.a 1 560.bf odd 4 1
3920.1.bt.a 2 112.v even 12 2
3920.1.bt.a 2 112.x odd 12 2
3920.1.bt.a 2 560.cn odd 12 2
3920.1.bt.a 2 560.co even 12 2
3920.1.bt.b 2 112.u odd 12 2
3920.1.bt.b 2 112.w even 12 2
3920.1.bt.b 2 560.cr even 12 2
3920.1.bt.b 2 560.cs odd 12 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(1280, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 1$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2}$$
$17$ $$T^{2}$$
$19$ $$T^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2} + 4$$
$31$ $$T^{2}$$
$37$ $$T^{2}$$
$41$ $$(T - 2)^{2}$$
$43$ $$T^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$T^{2} + 4$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$T^{2}$$
$79$ $$T^{2}$$
$83$ $$T^{2}$$
$89$ $$(T + 2)^{2}$$
$97$ $$T^{2}$$