# Properties

 Label 1280.2.d.j Level $1280$ Weight $2$ Character orbit 1280.d Analytic conductor $10.221$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1280,2,Mod(641,1280)]

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([0, 1, 0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1280.641");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1280 = 2^{8} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1280.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: $$10.2208514587$$ 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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 40) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + i q^{5} + 4 q^{7} + 3 q^{9}+O(q^{10})$$ q + i * q^5 + 4 * q^7 + 3 * q^9 $$q + i q^{5} + 4 q^{7} + 3 q^{9} + 4 i q^{11} + 2 i q^{13} + 2 q^{17} - 4 i q^{19} - 4 q^{23} - q^{25} + 2 i q^{29} - 8 q^{31} + 4 i q^{35} + 6 i q^{37} + 6 q^{41} - 8 i q^{43} + 3 i q^{45} + 4 q^{47} + 9 q^{49} + 6 i q^{53} - 4 q^{55} - 4 i q^{59} + 2 i q^{61} + 12 q^{63} - 2 q^{65} - 8 i q^{67} + 6 q^{73} + 16 i q^{77} + 9 q^{81} + 16 i q^{83} + 2 i q^{85} + 6 q^{89} + 8 i q^{91} + 4 q^{95} - 14 q^{97} + 12 i q^{99} +O(q^{100})$$ q + i * q^5 + 4 * q^7 + 3 * q^9 + 4*i * q^11 + 2*i * q^13 + 2 * q^17 - 4*i * q^19 - 4 * q^23 - q^25 + 2*i * q^29 - 8 * q^31 + 4*i * q^35 + 6*i * q^37 + 6 * q^41 - 8*i * q^43 + 3*i * q^45 + 4 * q^47 + 9 * q^49 + 6*i * q^53 - 4 * q^55 - 4*i * q^59 + 2*i * q^61 + 12 * q^63 - 2 * q^65 - 8*i * q^67 + 6 * q^73 + 16*i * q^77 + 9 * q^81 + 16*i * q^83 + 2*i * q^85 + 6 * q^89 + 8*i * q^91 + 4 * q^95 - 14 * q^97 + 12*i * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 8 q^{7} + 6 q^{9}+O(q^{10})$$ 2 * q + 8 * q^7 + 6 * q^9 $$2 q + 8 q^{7} + 6 q^{9} + 4 q^{17} - 8 q^{23} - 2 q^{25} - 16 q^{31} + 12 q^{41} + 8 q^{47} + 18 q^{49} - 8 q^{55} + 24 q^{63} - 4 q^{65} + 12 q^{73} + 18 q^{81} + 12 q^{89} + 8 q^{95} - 28 q^{97}+O(q^{100})$$ 2 * q + 8 * q^7 + 6 * q^9 + 4 * q^17 - 8 * q^23 - 2 * q^25 - 16 * q^31 + 12 * q^41 + 8 * q^47 + 18 * q^49 - 8 * q^55 + 24 * q^63 - 4 * q^65 + 12 * q^73 + 18 * q^81 + 12 * q^89 + 8 * q^95 - 28 * q^97

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
641.1
 − 1.00000i 1.00000i
0 0 0 1.00000i 0 4.00000 0 3.00000 0
641.2 0 0 0 1.00000i 0 4.00000 0 3.00000 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
8.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1280.2.d.j 2
4.b odd 2 1 1280.2.d.a 2
8.b even 2 1 inner 1280.2.d.j 2
8.d odd 2 1 1280.2.d.a 2
16.e even 4 1 40.2.a.a 1
16.e even 4 1 320.2.a.c 1
16.f odd 4 1 80.2.a.a 1
16.f odd 4 1 320.2.a.d 1
48.i odd 4 1 360.2.a.a 1
48.i odd 4 1 2880.2.a.t 1
48.k even 4 1 720.2.a.e 1
48.k even 4 1 2880.2.a.bg 1
80.i odd 4 1 200.2.c.b 2
80.i odd 4 1 1600.2.c.k 2
80.j even 4 1 400.2.c.d 2
80.j even 4 1 1600.2.c.m 2
80.k odd 4 1 400.2.a.e 1
80.k odd 4 1 1600.2.a.k 1
80.q even 4 1 200.2.a.c 1
80.q even 4 1 1600.2.a.o 1
80.s even 4 1 400.2.c.d 2
80.s even 4 1 1600.2.c.m 2
80.t odd 4 1 200.2.c.b 2
80.t odd 4 1 1600.2.c.k 2
112.j even 4 1 3920.2.a.s 1
112.l odd 4 1 1960.2.a.g 1
112.w even 12 2 1960.2.q.h 2
112.x odd 12 2 1960.2.q.i 2
144.w odd 12 2 3240.2.q.x 2
144.x even 12 2 3240.2.q.k 2
176.i even 4 1 9680.2.a.q 1
176.l odd 4 1 4840.2.a.f 1
208.p even 4 1 6760.2.a.i 1
240.t even 4 1 3600.2.a.h 1
240.z odd 4 1 3600.2.f.t 2
240.bb even 4 1 1800.2.f.a 2
240.bd odd 4 1 3600.2.f.t 2
240.bf even 4 1 1800.2.f.a 2
240.bm odd 4 1 1800.2.a.v 1
560.bf odd 4 1 9800.2.a.x 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.2.a.a 1 16.e even 4 1
80.2.a.a 1 16.f odd 4 1
200.2.a.c 1 80.q even 4 1
200.2.c.b 2 80.i odd 4 1
200.2.c.b 2 80.t odd 4 1
320.2.a.c 1 16.e even 4 1
320.2.a.d 1 16.f odd 4 1
360.2.a.a 1 48.i odd 4 1
400.2.a.e 1 80.k odd 4 1
400.2.c.d 2 80.j even 4 1
400.2.c.d 2 80.s even 4 1
720.2.a.e 1 48.k even 4 1
1280.2.d.a 2 4.b odd 2 1
1280.2.d.a 2 8.d odd 2 1
1280.2.d.j 2 1.a even 1 1 trivial
1280.2.d.j 2 8.b even 2 1 inner
1600.2.a.k 1 80.k odd 4 1
1600.2.a.o 1 80.q even 4 1
1600.2.c.k 2 80.i odd 4 1
1600.2.c.k 2 80.t odd 4 1
1600.2.c.m 2 80.j even 4 1
1600.2.c.m 2 80.s even 4 1
1800.2.a.v 1 240.bm odd 4 1
1800.2.f.a 2 240.bb even 4 1
1800.2.f.a 2 240.bf even 4 1
1960.2.a.g 1 112.l odd 4 1
1960.2.q.h 2 112.w even 12 2
1960.2.q.i 2 112.x odd 12 2
2880.2.a.t 1 48.i odd 4 1
2880.2.a.bg 1 48.k even 4 1
3240.2.q.k 2 144.x even 12 2
3240.2.q.x 2 144.w odd 12 2
3600.2.a.h 1 240.t even 4 1
3600.2.f.t 2 240.z odd 4 1
3600.2.f.t 2 240.bd odd 4 1
3920.2.a.s 1 112.j even 4 1
4840.2.a.f 1 176.l odd 4 1
6760.2.a.i 1 208.p even 4 1
9680.2.a.q 1 176.i even 4 1
9800.2.a.x 1 560.bf odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1280, [\chi])$$:

 $$T_{3}$$ T3 $$T_{7} - 4$$ T7 - 4 $$T_{11}^{2} + 16$$ T11^2 + 16 $$T_{31} + 8$$ T31 + 8

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 1$$
$7$ $$(T - 4)^{2}$$
$11$ $$T^{2} + 16$$
$13$ $$T^{2} + 4$$
$17$ $$(T - 2)^{2}$$
$19$ $$T^{2} + 16$$
$23$ $$(T + 4)^{2}$$
$29$ $$T^{2} + 4$$
$31$ $$(T + 8)^{2}$$
$37$ $$T^{2} + 36$$
$41$ $$(T - 6)^{2}$$
$43$ $$T^{2} + 64$$
$47$ $$(T - 4)^{2}$$
$53$ $$T^{2} + 36$$
$59$ $$T^{2} + 16$$
$61$ $$T^{2} + 4$$
$67$ $$T^{2} + 64$$
$71$ $$T^{2}$$
$73$ $$(T - 6)^{2}$$
$79$ $$T^{2}$$
$83$ $$T^{2} + 256$$
$89$ $$(T - 6)^{2}$$
$97$ $$(T + 14)^{2}$$