# Properties

 Label 1280.1.h.c Level $1280$ Weight $1$ Character orbit 1280.h Self dual yes Analytic conductor $0.639$ Analytic rank $0$ Dimension $2$ Projective image $D_{4}$ CM discriminant -20 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1280,1,Mod(1279,1280)]

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, 0, 1]))

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

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

chi := DirichletCharacter("1280.1279");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1280 = 2^{8} \cdot 5$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1280.h (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: yes Analytic conductor: $$0.638803216170$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 640) Projective image: $$D_{4}$$ Projective field: Galois closure of 4.2.1600.1 Artin image: $D_8$ Artin field: Galois closure of 8.0.2097152000.6

## $q$-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 $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta q^{3} + q^{5} + \beta q^{7} + q^{9} +O(q^{10})$$ q - b * q^3 + q^5 + b * q^7 + q^9 $$q - \beta q^{3} + q^{5} + \beta q^{7} + q^{9} - \beta q^{15} - 2 q^{21} - \beta q^{23} + q^{25} + \beta q^{35} + \beta q^{43} + q^{45} + \beta q^{47} + q^{49} - 2 q^{61} + \beta q^{63} + \beta q^{67} + 2 q^{69} - \beta q^{75} - q^{81} + \beta q^{83} - 2 q^{89} +O(q^{100})$$ q - b * q^3 + q^5 + b * q^7 + q^9 - b * q^15 - 2 * q^21 - b * q^23 + q^25 + b * q^35 + b * q^43 + q^45 + b * q^47 + q^49 - 2 * q^61 + b * q^63 + b * q^67 + 2 * q^69 - b * q^75 - q^81 + b * q^83 - 2 * q^89 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{5} + 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^5 + 2 * q^9 $$2 q + 2 q^{5} + 2 q^{9} - 4 q^{21} + 2 q^{25} + 2 q^{45} + 2 q^{49} - 4 q^{61} + 4 q^{69} - 2 q^{81} - 4 q^{89}+O(q^{100})$$ 2 * q + 2 * q^5 + 2 * q^9 - 4 * q^21 + 2 * q^25 + 2 * q^45 + 2 * q^49 - 4 * q^61 + 4 * q^69 - 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1279.1
 1.41421 −1.41421
0 −1.41421 0 1.00000 0 1.41421 0 1.00000 0
1279.2 0 1.41421 0 1.00000 0 −1.41421 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
20.d odd 2 1 CM by $$\Q(\sqrt{-5})$$
4.b odd 2 1 inner
5.b 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.h.c 2
4.b odd 2 1 inner 1280.1.h.c 2
5.b even 2 1 inner 1280.1.h.c 2
8.b even 2 1 1280.1.h.a 2
8.d odd 2 1 1280.1.h.a 2
16.e even 4 2 640.1.e.c 4
16.f odd 4 2 640.1.e.c 4
20.d odd 2 1 CM 1280.1.h.c 2
40.e odd 2 1 1280.1.h.a 2
40.f even 2 1 1280.1.h.a 2
80.i odd 4 2 3200.1.g.e 4
80.j even 4 2 3200.1.g.e 4
80.k odd 4 2 640.1.e.c 4
80.q even 4 2 640.1.e.c 4
80.s even 4 2 3200.1.g.e 4
80.t odd 4 2 3200.1.g.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
640.1.e.c 4 16.e even 4 2
640.1.e.c 4 16.f odd 4 2
640.1.e.c 4 80.k odd 4 2
640.1.e.c 4 80.q even 4 2
1280.1.h.a 2 8.b even 2 1
1280.1.h.a 2 8.d odd 2 1
1280.1.h.a 2 40.e odd 2 1
1280.1.h.a 2 40.f even 2 1
1280.1.h.c 2 1.a even 1 1 trivial
1280.1.h.c 2 4.b odd 2 1 inner
1280.1.h.c 2 5.b even 2 1 inner
1280.1.h.c 2 20.d odd 2 1 CM
3200.1.g.e 4 80.i odd 4 2
3200.1.g.e 4 80.j even 4 2
3200.1.g.e 4 80.s even 4 2
3200.1.g.e 4 80.t odd 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(1280, [\chi])$$:

 $$T_{3}^{2} - 2$$ T3^2 - 2 $$T_{61} + 2$$ T61 + 2

## Hecke characteristic polynomials

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