# Properties

 Label 1280.4.d.u 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: $$\Q(i, \sqrt{10})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 25$$ x^4 + 25 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{3}$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{3} - 5 \beta_1 q^{5} - 3 \beta_{3} q^{7} - 13 q^{9}+O(q^{10})$$ q + b2 * q^3 - 5*b1 * q^5 - 3*b3 * q^7 - 13 * q^9 $$q + \beta_{2} q^{3} - 5 \beta_1 q^{5} - 3 \beta_{3} q^{7} - 13 q^{9} + 2 \beta_{2} q^{11} + 38 \beta_1 q^{13} + 5 \beta_{3} q^{15} + 34 q^{17} - 16 \beta_{2} q^{19} - 120 \beta_1 q^{21} - 13 \beta_{3} q^{23} - 25 q^{25} + 14 \beta_{2} q^{27} + 270 \beta_1 q^{29} - 54 \beta_{3} q^{31} - 80 q^{33} + 15 \beta_{2} q^{35} - 206 \beta_1 q^{37} - 38 \beta_{3} q^{39} + 270 q^{41} - 85 \beta_{2} q^{43} + 65 \beta_1 q^{45} - 21 \beta_{3} q^{47} + 17 q^{49} + 34 \beta_{2} q^{51} + 258 \beta_1 q^{53} + 10 \beta_{3} q^{55} + 640 q^{57} + 12 \beta_{2} q^{59} - 250 \beta_1 q^{61} + 39 \beta_{3} q^{63} + 190 q^{65} - 129 \beta_{2} q^{67} - 520 \beta_1 q^{69} + 102 \beta_{3} q^{71} + 1078 q^{73} - 25 \beta_{2} q^{75} - 240 \beta_1 q^{77} - 44 \beta_{3} q^{79} - 911 q^{81} - 175 \beta_{2} q^{83} - 170 \beta_1 q^{85} - 270 \beta_{3} q^{87} - 890 q^{89} - 114 \beta_{2} q^{91} - 2160 \beta_1 q^{93} - 80 \beta_{3} q^{95} - 254 q^{97} - 26 \beta_{2} q^{99}+O(q^{100})$$ q + b2 * q^3 - 5*b1 * q^5 - 3*b3 * q^7 - 13 * q^9 + 2*b2 * q^11 + 38*b1 * q^13 + 5*b3 * q^15 + 34 * q^17 - 16*b2 * q^19 - 120*b1 * q^21 - 13*b3 * q^23 - 25 * q^25 + 14*b2 * q^27 + 270*b1 * q^29 - 54*b3 * q^31 - 80 * q^33 + 15*b2 * q^35 - 206*b1 * q^37 - 38*b3 * q^39 + 270 * q^41 - 85*b2 * q^43 + 65*b1 * q^45 - 21*b3 * q^47 + 17 * q^49 + 34*b2 * q^51 + 258*b1 * q^53 + 10*b3 * q^55 + 640 * q^57 + 12*b2 * q^59 - 250*b1 * q^61 + 39*b3 * q^63 + 190 * q^65 - 129*b2 * q^67 - 520*b1 * q^69 + 102*b3 * q^71 + 1078 * q^73 - 25*b2 * q^75 - 240*b1 * q^77 - 44*b3 * q^79 - 911 * q^81 - 175*b2 * q^83 - 170*b1 * q^85 - 270*b3 * q^87 - 890 * q^89 - 114*b2 * q^91 - 2160*b1 * q^93 - 80*b3 * q^95 - 254 * q^97 - 26*b2 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 52 q^{9}+O(q^{10})$$ 4 * q - 52 * q^9 $$4 q - 52 q^{9} + 136 q^{17} - 100 q^{25} - 320 q^{33} + 1080 q^{41} + 68 q^{49} + 2560 q^{57} + 760 q^{65} + 4312 q^{73} - 3644 q^{81} - 3560 q^{89} - 1016 q^{97}+O(q^{100})$$ 4 * q - 52 * q^9 + 136 * q^17 - 100 * q^25 - 320 * q^33 + 1080 * q^41 + 68 * q^49 + 2560 * q^57 + 760 * q^65 + 4312 * q^73 - 3644 * q^81 - 3560 * q^89 - 1016 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 25$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{2} ) / 5$$ (v^2) / 5 $$\beta_{2}$$ $$=$$ $$( 2\nu^{3} + 10\nu ) / 5$$ (2*v^3 + 10*v) / 5 $$\beta_{3}$$ $$=$$ $$( -2\nu^{3} + 10\nu ) / 5$$ (-2*v^3 + 10*v) / 5
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_{2} ) / 4$$ (b3 + b2) / 4 $$\nu^{2}$$ $$=$$ $$5\beta_1$$ 5*b1 $$\nu^{3}$$ $$=$$ $$( -5\beta_{3} + 5\beta_{2} ) / 4$$ (-5*b3 + 5*b2) / 4

## 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
 −1.58114 − 1.58114i 1.58114 − 1.58114i 1.58114 + 1.58114i −1.58114 + 1.58114i
0 6.32456i 0 5.00000i 0 18.9737 0 −13.0000 0
641.2 0 6.32456i 0 5.00000i 0 −18.9737 0 −13.0000 0
641.3 0 6.32456i 0 5.00000i 0 −18.9737 0 −13.0000 0
641.4 0 6.32456i 0 5.00000i 0 18.9737 0 −13.0000 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
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.u 4
4.b odd 2 1 inner 1280.4.d.u 4
8.b even 2 1 inner 1280.4.d.u 4
8.d odd 2 1 inner 1280.4.d.u 4
16.e even 4 1 160.4.a.f 2
16.e even 4 1 320.4.a.p 2
16.f odd 4 1 160.4.a.f 2
16.f odd 4 1 320.4.a.p 2
48.i odd 4 1 1440.4.a.v 2
48.k even 4 1 1440.4.a.v 2
80.i odd 4 1 800.4.c.j 4
80.j even 4 1 800.4.c.j 4
80.k odd 4 1 800.4.a.p 2
80.k odd 4 1 1600.4.a.ch 2
80.q even 4 1 800.4.a.p 2
80.q even 4 1 1600.4.a.ch 2
80.s even 4 1 800.4.c.j 4
80.t odd 4 1 800.4.c.j 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.4.a.f 2 16.e even 4 1
160.4.a.f 2 16.f odd 4 1
320.4.a.p 2 16.e even 4 1
320.4.a.p 2 16.f odd 4 1
800.4.a.p 2 80.k odd 4 1
800.4.a.p 2 80.q even 4 1
800.4.c.j 4 80.i odd 4 1
800.4.c.j 4 80.j even 4 1
800.4.c.j 4 80.s even 4 1
800.4.c.j 4 80.t odd 4 1
1280.4.d.u 4 1.a even 1 1 trivial
1280.4.d.u 4 4.b odd 2 1 inner
1280.4.d.u 4 8.b even 2 1 inner
1280.4.d.u 4 8.d odd 2 1 inner
1440.4.a.v 2 48.i odd 4 1
1440.4.a.v 2 48.k even 4 1
1600.4.a.ch 2 80.k odd 4 1
1600.4.a.ch 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 $$S_{4}^{\mathrm{new}}(1280, [\chi])$$:

 $$T_{3}^{2} + 40$$ T3^2 + 40 $$T_{7}^{2} - 360$$ T7^2 - 360 $$T_{11}^{2} + 160$$ T11^2 + 160

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T^{2} + 40)^{2}$$
$5$ $$(T^{2} + 25)^{2}$$
$7$ $$(T^{2} - 360)^{2}$$
$11$ $$(T^{2} + 160)^{2}$$
$13$ $$(T^{2} + 1444)^{2}$$
$17$ $$(T - 34)^{4}$$
$19$ $$(T^{2} + 10240)^{2}$$
$23$ $$(T^{2} - 6760)^{2}$$
$29$ $$(T^{2} + 72900)^{2}$$
$31$ $$(T^{2} - 116640)^{2}$$
$37$ $$(T^{2} + 42436)^{2}$$
$41$ $$(T - 270)^{4}$$
$43$ $$(T^{2} + 289000)^{2}$$
$47$ $$(T^{2} - 17640)^{2}$$
$53$ $$(T^{2} + 66564)^{2}$$
$59$ $$(T^{2} + 5760)^{2}$$
$61$ $$(T^{2} + 62500)^{2}$$
$67$ $$(T^{2} + 665640)^{2}$$
$71$ $$(T^{2} - 416160)^{2}$$
$73$ $$(T - 1078)^{4}$$
$79$ $$(T^{2} - 77440)^{2}$$
$83$ $$(T^{2} + 1225000)^{2}$$
$89$ $$(T + 890)^{4}$$
$97$ $$(T + 254)^{4}$$