# Properties

 Label 1280.2.d.c Level $1280$ Weight $2$ Character orbit 1280.d Analytic conductor $10.221$ Analytic rank $1$ 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: $$1$$ 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 20) 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.

 $$f(q)$$ $$=$$ $$q + 2 i q^{3} - i q^{5} - 2 q^{7} - q^{9} +O(q^{10})$$ q + 2*i * q^3 - i * q^5 - 2 * q^7 - q^9 $$q + 2 i q^{3} - i q^{5} - 2 q^{7} - q^{9} - 2 i q^{13} + 2 q^{15} - 6 q^{17} + 4 i q^{19} - 4 i q^{21} - 6 q^{23} - q^{25} + 4 i q^{27} - 6 i q^{29} - 4 q^{31} + 2 i q^{35} + 2 i q^{37} + 4 q^{39} - 6 q^{41} - 10 i q^{43} + i q^{45} - 6 q^{47} - 3 q^{49} - 12 i q^{51} - 6 i q^{53} - 8 q^{57} + 12 i q^{59} - 2 i q^{61} + 2 q^{63} - 2 q^{65} - 2 i q^{67} - 12 i q^{69} + 12 q^{71} - 2 q^{73} - 2 i q^{75} + 8 q^{79} - 11 q^{81} - 6 i q^{83} + 6 i q^{85} + 12 q^{87} + 6 q^{89} + 4 i q^{91} - 8 i q^{93} + 4 q^{95} + 2 q^{97} +O(q^{100})$$ q + 2*i * q^3 - i * q^5 - 2 * q^7 - q^9 - 2*i * q^13 + 2 * q^15 - 6 * q^17 + 4*i * q^19 - 4*i * q^21 - 6 * q^23 - q^25 + 4*i * q^27 - 6*i * q^29 - 4 * q^31 + 2*i * q^35 + 2*i * q^37 + 4 * q^39 - 6 * q^41 - 10*i * q^43 + i * q^45 - 6 * q^47 - 3 * q^49 - 12*i * q^51 - 6*i * q^53 - 8 * q^57 + 12*i * q^59 - 2*i * q^61 + 2 * q^63 - 2 * q^65 - 2*i * q^67 - 12*i * q^69 + 12 * q^71 - 2 * q^73 - 2*i * q^75 + 8 * q^79 - 11 * q^81 - 6*i * q^83 + 6*i * q^85 + 12 * q^87 + 6 * q^89 + 4*i * q^91 - 8*i * q^93 + 4 * q^95 + 2 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{7} - 2 q^{9}+O(q^{10})$$ 2 * q - 4 * q^7 - 2 * q^9 $$2 q - 4 q^{7} - 2 q^{9} + 4 q^{15} - 12 q^{17} - 12 q^{23} - 2 q^{25} - 8 q^{31} + 8 q^{39} - 12 q^{41} - 12 q^{47} - 6 q^{49} - 16 q^{57} + 4 q^{63} - 4 q^{65} + 24 q^{71} - 4 q^{73} + 16 q^{79} - 22 q^{81} + 24 q^{87} + 12 q^{89} + 8 q^{95} + 4 q^{97}+O(q^{100})$$ 2 * q - 4 * q^7 - 2 * q^9 + 4 * q^15 - 12 * q^17 - 12 * q^23 - 2 * q^25 - 8 * q^31 + 8 * q^39 - 12 * q^41 - 12 * q^47 - 6 * q^49 - 16 * q^57 + 4 * q^63 - 4 * q^65 + 24 * q^71 - 4 * q^73 + 16 * q^79 - 22 * q^81 + 24 * q^87 + 12 * q^89 + 8 * q^95 + 4 * 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 2.00000i 0 1.00000i 0 −2.00000 0 −1.00000 0
641.2 0 2.00000i 0 1.00000i 0 −2.00000 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
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.c 2
4.b odd 2 1 1280.2.d.g 2
8.b even 2 1 inner 1280.2.d.c 2
8.d odd 2 1 1280.2.d.g 2
16.e even 4 1 20.2.a.a 1
16.e even 4 1 320.2.a.f 1
16.f odd 4 1 80.2.a.b 1
16.f odd 4 1 320.2.a.a 1
48.i odd 4 1 180.2.a.a 1
48.i odd 4 1 2880.2.a.m 1
48.k even 4 1 720.2.a.h 1
48.k even 4 1 2880.2.a.f 1
80.i odd 4 1 100.2.c.a 2
80.i odd 4 1 1600.2.c.d 2
80.j even 4 1 400.2.c.b 2
80.j even 4 1 1600.2.c.e 2
80.k odd 4 1 400.2.a.c 1
80.k odd 4 1 1600.2.a.w 1
80.q even 4 1 100.2.a.a 1
80.q even 4 1 1600.2.a.c 1
80.s even 4 1 400.2.c.b 2
80.s even 4 1 1600.2.c.e 2
80.t odd 4 1 100.2.c.a 2
80.t odd 4 1 1600.2.c.d 2
112.j even 4 1 3920.2.a.h 1
112.l odd 4 1 980.2.a.h 1
112.w even 12 2 980.2.i.i 2
112.x odd 12 2 980.2.i.c 2
144.w odd 12 2 1620.2.i.b 2
144.x even 12 2 1620.2.i.h 2
176.i even 4 1 9680.2.a.ba 1
176.l odd 4 1 2420.2.a.a 1
208.m odd 4 1 3380.2.f.b 2
208.p even 4 1 3380.2.a.c 1
208.r odd 4 1 3380.2.f.b 2
240.t even 4 1 3600.2.a.be 1
240.z odd 4 1 3600.2.f.j 2
240.bb even 4 1 900.2.d.c 2
240.bd odd 4 1 3600.2.f.j 2
240.bf even 4 1 900.2.d.c 2
240.bm odd 4 1 900.2.a.b 1
272.j even 4 1 5780.2.c.a 2
272.r even 4 1 5780.2.a.f 1
272.s even 4 1 5780.2.c.a 2
304.j odd 4 1 7220.2.a.f 1
336.y even 4 1 8820.2.a.g 1
560.r even 4 1 4900.2.e.f 2
560.bf odd 4 1 4900.2.a.e 1
560.bn even 4 1 4900.2.e.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.2.a.a 1 16.e even 4 1
80.2.a.b 1 16.f odd 4 1
100.2.a.a 1 80.q even 4 1
100.2.c.a 2 80.i odd 4 1
100.2.c.a 2 80.t odd 4 1
180.2.a.a 1 48.i odd 4 1
320.2.a.a 1 16.f odd 4 1
320.2.a.f 1 16.e even 4 1
400.2.a.c 1 80.k odd 4 1
400.2.c.b 2 80.j even 4 1
400.2.c.b 2 80.s even 4 1
720.2.a.h 1 48.k even 4 1
900.2.a.b 1 240.bm odd 4 1
900.2.d.c 2 240.bb even 4 1
900.2.d.c 2 240.bf even 4 1
980.2.a.h 1 112.l odd 4 1
980.2.i.c 2 112.x odd 12 2
980.2.i.i 2 112.w even 12 2
1280.2.d.c 2 1.a even 1 1 trivial
1280.2.d.c 2 8.b even 2 1 inner
1280.2.d.g 2 4.b odd 2 1
1280.2.d.g 2 8.d odd 2 1
1600.2.a.c 1 80.q even 4 1
1600.2.a.w 1 80.k odd 4 1
1600.2.c.d 2 80.i odd 4 1
1600.2.c.d 2 80.t odd 4 1
1600.2.c.e 2 80.j even 4 1
1600.2.c.e 2 80.s even 4 1
1620.2.i.b 2 144.w odd 12 2
1620.2.i.h 2 144.x even 12 2
2420.2.a.a 1 176.l odd 4 1
2880.2.a.f 1 48.k even 4 1
2880.2.a.m 1 48.i odd 4 1
3380.2.a.c 1 208.p even 4 1
3380.2.f.b 2 208.m odd 4 1
3380.2.f.b 2 208.r odd 4 1
3600.2.a.be 1 240.t even 4 1
3600.2.f.j 2 240.z odd 4 1
3600.2.f.j 2 240.bd odd 4 1
3920.2.a.h 1 112.j even 4 1
4900.2.a.e 1 560.bf odd 4 1
4900.2.e.f 2 560.r even 4 1
4900.2.e.f 2 560.bn even 4 1
5780.2.a.f 1 272.r even 4 1
5780.2.c.a 2 272.j even 4 1
5780.2.c.a 2 272.s even 4 1
7220.2.a.f 1 304.j odd 4 1
8820.2.a.g 1 336.y even 4 1
9680.2.a.ba 1 176.i even 4 1

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 4$$
$5$ $$T^{2} + 1$$
$7$ $$(T + 2)^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 4$$
$17$ $$(T + 6)^{2}$$
$19$ $$T^{2} + 16$$
$23$ $$(T + 6)^{2}$$
$29$ $$T^{2} + 36$$
$31$ $$(T + 4)^{2}$$
$37$ $$T^{2} + 4$$
$41$ $$(T + 6)^{2}$$
$43$ $$T^{2} + 100$$
$47$ $$(T + 6)^{2}$$
$53$ $$T^{2} + 36$$
$59$ $$T^{2} + 144$$
$61$ $$T^{2} + 4$$
$67$ $$T^{2} + 4$$
$71$ $$(T - 12)^{2}$$
$73$ $$(T + 2)^{2}$$
$79$ $$(T - 8)^{2}$$
$83$ $$T^{2} + 36$$
$89$ $$(T - 6)^{2}$$
$97$ $$(T - 2)^{2}$$