# Properties

 Label 1280.4.d.f Level $1280$ Weight $4$ Character orbit 1280.d Analytic conductor $75.522$ 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] = [1280,4,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, 4, names="a")

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

chi := DirichletCharacter("1280.641");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1280 = 2^{8} \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ 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: $$75.5224448073$$ 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 160) 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.

 $$f(q)$$ $$=$$ $$q + 2 i q^{3} - 5 i q^{5} - 6 q^{7} + 23 q^{9} +O(q^{10})$$ q + 2*i * q^3 - 5*i * q^5 - 6 * q^7 + 23 * q^9 $$q + 2 i q^{3} - 5 i q^{5} - 6 q^{7} + 23 q^{9} + 60 i q^{11} - 50 i q^{13} + 10 q^{15} - 30 q^{17} - 40 i q^{19} - 12 i q^{21} - 178 q^{23} - 25 q^{25} + 100 i q^{27} - 166 i q^{29} + 20 q^{31} - 120 q^{33} + 30 i q^{35} + 10 i q^{37} + 100 q^{39} + 250 q^{41} + 142 i q^{43} - 115 i q^{45} + 214 q^{47} - 307 q^{49} - 60 i q^{51} + 490 i q^{53} + 300 q^{55} + 80 q^{57} - 800 i q^{59} - 250 i q^{61} - 138 q^{63} - 250 q^{65} + 774 i q^{67} - 356 i q^{69} - 100 q^{71} + 230 q^{73} - 50 i q^{75} - 360 i q^{77} - 1320 q^{79} + 421 q^{81} - 982 i q^{83} + 150 i q^{85} + 332 q^{87} - 874 q^{89} + 300 i q^{91} + 40 i q^{93} - 200 q^{95} - 310 q^{97} + 1380 i q^{99} +O(q^{100})$$ q + 2*i * q^3 - 5*i * q^5 - 6 * q^7 + 23 * q^9 + 60*i * q^11 - 50*i * q^13 + 10 * q^15 - 30 * q^17 - 40*i * q^19 - 12*i * q^21 - 178 * q^23 - 25 * q^25 + 100*i * q^27 - 166*i * q^29 + 20 * q^31 - 120 * q^33 + 30*i * q^35 + 10*i * q^37 + 100 * q^39 + 250 * q^41 + 142*i * q^43 - 115*i * q^45 + 214 * q^47 - 307 * q^49 - 60*i * q^51 + 490*i * q^53 + 300 * q^55 + 80 * q^57 - 800*i * q^59 - 250*i * q^61 - 138 * q^63 - 250 * q^65 + 774*i * q^67 - 356*i * q^69 - 100 * q^71 + 230 * q^73 - 50*i * q^75 - 360*i * q^77 - 1320 * q^79 + 421 * q^81 - 982*i * q^83 + 150*i * q^85 + 332 * q^87 - 874 * q^89 + 300*i * q^91 + 40*i * q^93 - 200 * q^95 - 310 * q^97 + 1380*i * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 12 q^{7} + 46 q^{9}+O(q^{10})$$ 2 * q - 12 * q^7 + 46 * q^9 $$2 q - 12 q^{7} + 46 q^{9} + 20 q^{15} - 60 q^{17} - 356 q^{23} - 50 q^{25} + 40 q^{31} - 240 q^{33} + 200 q^{39} + 500 q^{41} + 428 q^{47} - 614 q^{49} + 600 q^{55} + 160 q^{57} - 276 q^{63} - 500 q^{65} - 200 q^{71} + 460 q^{73} - 2640 q^{79} + 842 q^{81} + 664 q^{87} - 1748 q^{89} - 400 q^{95} - 620 q^{97}+O(q^{100})$$ 2 * q - 12 * q^7 + 46 * q^9 + 20 * q^15 - 60 * q^17 - 356 * q^23 - 50 * q^25 + 40 * q^31 - 240 * q^33 + 200 * q^39 + 500 * q^41 + 428 * q^47 - 614 * q^49 + 600 * q^55 + 160 * q^57 - 276 * q^63 - 500 * q^65 - 200 * q^71 + 460 * q^73 - 2640 * q^79 + 842 * q^81 + 664 * q^87 - 1748 * q^89 - 400 * q^95 - 620 * 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 5.00000i 0 −6.00000 0 23.0000 0
641.2 0 2.00000i 0 5.00000i 0 −6.00000 0 23.0000 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 1280.4.d.f 2
4.b odd 2 1 1280.4.d.k 2
8.b even 2 1 inner 1280.4.d.f 2
8.d odd 2 1 1280.4.d.k 2
16.e even 4 1 160.4.a.a 1
16.e even 4 1 320.4.a.i 1
16.f odd 4 1 160.4.a.b yes 1
16.f odd 4 1 320.4.a.f 1
48.i odd 4 1 1440.4.a.o 1
48.k even 4 1 1440.4.a.n 1
80.i odd 4 1 800.4.c.f 2
80.j even 4 1 800.4.c.e 2
80.k odd 4 1 800.4.a.d 1
80.k odd 4 1 1600.4.a.bj 1
80.q even 4 1 800.4.a.h 1
80.q even 4 1 1600.4.a.r 1
80.s even 4 1 800.4.c.e 2
80.t odd 4 1 800.4.c.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.4.a.a 1 16.e even 4 1
160.4.a.b yes 1 16.f odd 4 1
320.4.a.f 1 16.f odd 4 1
320.4.a.i 1 16.e even 4 1
800.4.a.d 1 80.k odd 4 1
800.4.a.h 1 80.q even 4 1
800.4.c.e 2 80.j even 4 1
800.4.c.e 2 80.s even 4 1
800.4.c.f 2 80.i odd 4 1
800.4.c.f 2 80.t odd 4 1
1280.4.d.f 2 1.a even 1 1 trivial
1280.4.d.f 2 8.b even 2 1 inner
1280.4.d.k 2 4.b odd 2 1
1280.4.d.k 2 8.d odd 2 1
1440.4.a.n 1 48.k even 4 1
1440.4.a.o 1 48.i odd 4 1
1600.4.a.r 1 80.q even 4 1
1600.4.a.bj 1 80.k odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(1280, [\chi])$$:

 $$T_{3}^{2} + 4$$ T3^2 + 4 $$T_{7} + 6$$ T7 + 6 $$T_{11}^{2} + 3600$$ T11^2 + 3600

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 4$$
$5$ $$T^{2} + 25$$
$7$ $$(T + 6)^{2}$$
$11$ $$T^{2} + 3600$$
$13$ $$T^{2} + 2500$$
$17$ $$(T + 30)^{2}$$
$19$ $$T^{2} + 1600$$
$23$ $$(T + 178)^{2}$$
$29$ $$T^{2} + 27556$$
$31$ $$(T - 20)^{2}$$
$37$ $$T^{2} + 100$$
$41$ $$(T - 250)^{2}$$
$43$ $$T^{2} + 20164$$
$47$ $$(T - 214)^{2}$$
$53$ $$T^{2} + 240100$$
$59$ $$T^{2} + 640000$$
$61$ $$T^{2} + 62500$$
$67$ $$T^{2} + 599076$$
$71$ $$(T + 100)^{2}$$
$73$ $$(T - 230)^{2}$$
$79$ $$(T + 1320)^{2}$$
$83$ $$T^{2} + 964324$$
$89$ $$(T + 874)^{2}$$
$97$ $$(T + 310)^{2}$$