# Properties

 Label 1280.4.d.v Level $1280$ Weight $4$ Character orbit 1280.d Analytic conductor $75.522$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# 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: $$4$$ Coefficient field: $$\Q(i, \sqrt{5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 3x^{2} + 1$$ x^4 + 3*x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{4}$$ 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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{3} + 5 \beta_1 q^{5} - 7 \beta_{3} q^{7} + 7 q^{9}+O(q^{10})$$ q - b2 * q^3 + 5*b1 * q^5 - 7*b3 * q^7 + 7 * q^9 $$q - \beta_{2} q^{3} + 5 \beta_1 q^{5} - 7 \beta_{3} q^{7} + 7 q^{9} - 2 \beta_{2} q^{11} - 62 \beta_1 q^{13} + 5 \beta_{3} q^{15} - 46 q^{17} - 24 \beta_{2} q^{19} + 140 \beta_1 q^{21} + 43 \beta_{3} q^{23} - 25 q^{25} - 34 \beta_{2} q^{27} - 90 \beta_1 q^{29} + 34 \beta_{3} q^{31} - 40 q^{33} - 35 \beta_{2} q^{35} + 214 \beta_1 q^{37} - 62 \beta_{3} q^{39} + 10 q^{41} - 15 \beta_{2} q^{43} + 35 \beta_1 q^{45} - 89 \beta_{3} q^{47} + 637 q^{49} + 46 \beta_{2} q^{51} + 678 \beta_1 q^{53} + 10 \beta_{3} q^{55} - 480 q^{57} - 92 \beta_{2} q^{59} + 250 \beta_1 q^{61} - 49 \beta_{3} q^{63} + 310 q^{65} - 11 \beta_{2} q^{67} - 860 \beta_1 q^{69} - 82 \beta_{3} q^{71} - 522 q^{73} + 25 \beta_{2} q^{75} + 280 \beta_1 q^{77} - 196 \beta_{3} q^{79} - 491 q^{81} - 85 \beta_{2} q^{83} - 230 \beta_1 q^{85} - 90 \beta_{3} q^{87} - 970 q^{89} + 434 \beta_{2} q^{91} - 680 \beta_1 q^{93} + 120 \beta_{3} q^{95} - 934 q^{97} - 14 \beta_{2} q^{99}+O(q^{100})$$ q - b2 * q^3 + 5*b1 * q^5 - 7*b3 * q^7 + 7 * q^9 - 2*b2 * q^11 - 62*b1 * q^13 + 5*b3 * q^15 - 46 * q^17 - 24*b2 * q^19 + 140*b1 * q^21 + 43*b3 * q^23 - 25 * q^25 - 34*b2 * q^27 - 90*b1 * q^29 + 34*b3 * q^31 - 40 * q^33 - 35*b2 * q^35 + 214*b1 * q^37 - 62*b3 * q^39 + 10 * q^41 - 15*b2 * q^43 + 35*b1 * q^45 - 89*b3 * q^47 + 637 * q^49 + 46*b2 * q^51 + 678*b1 * q^53 + 10*b3 * q^55 - 480 * q^57 - 92*b2 * q^59 + 250*b1 * q^61 - 49*b3 * q^63 + 310 * q^65 - 11*b2 * q^67 - 860*b1 * q^69 - 82*b3 * q^71 - 522 * q^73 + 25*b2 * q^75 + 280*b1 * q^77 - 196*b3 * q^79 - 491 * q^81 - 85*b2 * q^83 - 230*b1 * q^85 - 90*b3 * q^87 - 970 * q^89 + 434*b2 * q^91 - 680*b1 * q^93 + 120*b3 * q^95 - 934 * q^97 - 14*b2 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 28 q^{9}+O(q^{10})$$ 4 * q + 28 * q^9 $$4 q + 28 q^{9} - 184 q^{17} - 100 q^{25} - 160 q^{33} + 40 q^{41} + 2548 q^{49} - 1920 q^{57} + 1240 q^{65} - 2088 q^{73} - 1964 q^{81} - 3880 q^{89} - 3736 q^{97}+O(q^{100})$$ 4 * q + 28 * q^9 - 184 * q^17 - 100 * q^25 - 160 * q^33 + 40 * q^41 + 2548 * q^49 - 1920 * q^57 + 1240 * q^65 - 2088 * q^73 - 1964 * q^81 - 3880 * q^89 - 3736 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 3x^{2} + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu^{3} + 2\nu$$ v^3 + 2*v $$\beta_{2}$$ $$=$$ $$2\nu^{3} + 8\nu$$ 2*v^3 + 8*v $$\beta_{3}$$ $$=$$ $$4\nu^{2} + 6$$ 4*v^2 + 6
 $$\nu$$ $$=$$ $$( \beta_{2} - 2\beta_1 ) / 4$$ (b2 - 2*b1) / 4 $$\nu^{2}$$ $$=$$ $$( \beta_{3} - 6 ) / 4$$ (b3 - 6) / 4 $$\nu^{3}$$ $$=$$ $$( -\beta_{2} + 4\beta_1 ) / 2$$ (-b2 + 4*b1) / 2

## 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
 1.61803i 0.618034i − 0.618034i − 1.61803i
0 4.47214i 0 5.00000i 0 31.3050 0 7.00000 0
641.2 0 4.47214i 0 5.00000i 0 −31.3050 0 7.00000 0
641.3 0 4.47214i 0 5.00000i 0 −31.3050 0 7.00000 0
641.4 0 4.47214i 0 5.00000i 0 31.3050 0 7.00000 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 inner
8.b even 2 1 inner
8.d odd 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.v 4
4.b odd 2 1 inner 1280.4.d.v 4
8.b even 2 1 inner 1280.4.d.v 4
8.d odd 2 1 inner 1280.4.d.v 4
16.e even 4 1 160.4.a.d 2
16.e even 4 1 320.4.a.q 2
16.f odd 4 1 160.4.a.d 2
16.f odd 4 1 320.4.a.q 2
48.i odd 4 1 1440.4.a.bb 2
48.k even 4 1 1440.4.a.bb 2
80.i odd 4 1 800.4.c.l 4
80.j even 4 1 800.4.c.l 4
80.k odd 4 1 800.4.a.o 2
80.k odd 4 1 1600.4.a.cg 2
80.q even 4 1 800.4.a.o 2
80.q even 4 1 1600.4.a.cg 2
80.s even 4 1 800.4.c.l 4
80.t odd 4 1 800.4.c.l 4

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

## Hecke kernels

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

 $$T_{3}^{2} + 20$$ T3^2 + 20 $$T_{7}^{2} - 980$$ T7^2 - 980 $$T_{11}^{2} + 80$$ T11^2 + 80

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T^{2} + 20)^{2}$$
$5$ $$(T^{2} + 25)^{2}$$
$7$ $$(T^{2} - 980)^{2}$$
$11$ $$(T^{2} + 80)^{2}$$
$13$ $$(T^{2} + 3844)^{2}$$
$17$ $$(T + 46)^{4}$$
$19$ $$(T^{2} + 11520)^{2}$$
$23$ $$(T^{2} - 36980)^{2}$$
$29$ $$(T^{2} + 8100)^{2}$$
$31$ $$(T^{2} - 23120)^{2}$$
$37$ $$(T^{2} + 45796)^{2}$$
$41$ $$(T - 10)^{4}$$
$43$ $$(T^{2} + 4500)^{2}$$
$47$ $$(T^{2} - 158420)^{2}$$
$53$ $$(T^{2} + 459684)^{2}$$
$59$ $$(T^{2} + 169280)^{2}$$
$61$ $$(T^{2} + 62500)^{2}$$
$67$ $$(T^{2} + 2420)^{2}$$
$71$ $$(T^{2} - 134480)^{2}$$
$73$ $$(T + 522)^{4}$$
$79$ $$(T^{2} - 768320)^{2}$$
$83$ $$(T^{2} + 144500)^{2}$$
$89$ $$(T + 970)^{4}$$
$97$ $$(T + 934)^{4}$$