# Properties

 Label 1280.4.d.h 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 640) 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.

 $$f(q)$$ $$=$$ $$q + 8 i q^{3} - 5 i q^{5} - 2 q^{7} - 37 q^{9} +O(q^{10})$$ q + 8*i * q^3 - 5*i * q^5 - 2 * q^7 - 37 * q^9 $$q + 8 i q^{3} - 5 i q^{5} - 2 q^{7} - 37 q^{9} - 22 i q^{11} + 10 i q^{13} + 40 q^{15} + 10 q^{17} + 110 i q^{19} - 16 i q^{21} + 154 q^{23} - 25 q^{25} - 80 i q^{27} + 222 i q^{29} - 92 q^{31} + 176 q^{33} + 10 i q^{35} + 34 i q^{37} - 80 q^{39} - 398 q^{41} - 268 i q^{43} + 185 i q^{45} - 10 q^{47} - 339 q^{49} + 80 i q^{51} + 582 i q^{53} - 110 q^{55} - 880 q^{57} - 746 i q^{59} - 226 i q^{61} + 74 q^{63} + 50 q^{65} - 172 i q^{67} + 1232 i q^{69} - 928 q^{71} - 570 q^{73} - 200 i q^{75} + 44 i q^{77} + 64 q^{79} - 359 q^{81} - 864 i q^{83} - 50 i q^{85} - 1776 q^{87} + 874 q^{89} - 20 i q^{91} - 736 i q^{93} + 550 q^{95} + 306 q^{97} + 814 i q^{99} +O(q^{100})$$ q + 8*i * q^3 - 5*i * q^5 - 2 * q^7 - 37 * q^9 - 22*i * q^11 + 10*i * q^13 + 40 * q^15 + 10 * q^17 + 110*i * q^19 - 16*i * q^21 + 154 * q^23 - 25 * q^25 - 80*i * q^27 + 222*i * q^29 - 92 * q^31 + 176 * q^33 + 10*i * q^35 + 34*i * q^37 - 80 * q^39 - 398 * q^41 - 268*i * q^43 + 185*i * q^45 - 10 * q^47 - 339 * q^49 + 80*i * q^51 + 582*i * q^53 - 110 * q^55 - 880 * q^57 - 746*i * q^59 - 226*i * q^61 + 74 * q^63 + 50 * q^65 - 172*i * q^67 + 1232*i * q^69 - 928 * q^71 - 570 * q^73 - 200*i * q^75 + 44*i * q^77 + 64 * q^79 - 359 * q^81 - 864*i * q^83 - 50*i * q^85 - 1776 * q^87 + 874 * q^89 - 20*i * q^91 - 736*i * q^93 + 550 * q^95 + 306 * q^97 + 814*i * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{7} - 74 q^{9}+O(q^{10})$$ 2 * q - 4 * q^7 - 74 * q^9 $$2 q - 4 q^{7} - 74 q^{9} + 80 q^{15} + 20 q^{17} + 308 q^{23} - 50 q^{25} - 184 q^{31} + 352 q^{33} - 160 q^{39} - 796 q^{41} - 20 q^{47} - 678 q^{49} - 220 q^{55} - 1760 q^{57} + 148 q^{63} + 100 q^{65} - 1856 q^{71} - 1140 q^{73} + 128 q^{79} - 718 q^{81} - 3552 q^{87} + 1748 q^{89} + 1100 q^{95} + 612 q^{97}+O(q^{100})$$ 2 * q - 4 * q^7 - 74 * q^9 + 80 * q^15 + 20 * q^17 + 308 * q^23 - 50 * q^25 - 184 * q^31 + 352 * q^33 - 160 * q^39 - 796 * q^41 - 20 * q^47 - 678 * q^49 - 220 * q^55 - 1760 * q^57 + 148 * q^63 + 100 * q^65 - 1856 * q^71 - 1140 * q^73 + 128 * q^79 - 718 * q^81 - 3552 * q^87 + 1748 * q^89 + 1100 * q^95 + 612 * 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 8.00000i 0 5.00000i 0 −2.00000 0 −37.0000 0
641.2 0 8.00000i 0 5.00000i 0 −2.00000 0 −37.0000 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.h 2
4.b odd 2 1 1280.4.d.i 2
8.b even 2 1 inner 1280.4.d.h 2
8.d odd 2 1 1280.4.d.i 2
16.e even 4 1 640.4.a.a 1
16.e even 4 1 640.4.a.d yes 1
16.f odd 4 1 640.4.a.b yes 1
16.f odd 4 1 640.4.a.c yes 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
640.4.a.a 1 16.e even 4 1
640.4.a.b yes 1 16.f odd 4 1
640.4.a.c yes 1 16.f odd 4 1
640.4.a.d yes 1 16.e even 4 1
1280.4.d.h 2 1.a even 1 1 trivial
1280.4.d.h 2 8.b even 2 1 inner
1280.4.d.i 2 4.b odd 2 1
1280.4.d.i 2 8.d odd 2 1

## Hecke kernels

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

 $$T_{3}^{2} + 64$$ T3^2 + 64 $$T_{7} + 2$$ T7 + 2 $$T_{11}^{2} + 484$$ T11^2 + 484

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 64$$
$5$ $$T^{2} + 25$$
$7$ $$(T + 2)^{2}$$
$11$ $$T^{2} + 484$$
$13$ $$T^{2} + 100$$
$17$ $$(T - 10)^{2}$$
$19$ $$T^{2} + 12100$$
$23$ $$(T - 154)^{2}$$
$29$ $$T^{2} + 49284$$
$31$ $$(T + 92)^{2}$$
$37$ $$T^{2} + 1156$$
$41$ $$(T + 398)^{2}$$
$43$ $$T^{2} + 71824$$
$47$ $$(T + 10)^{2}$$
$53$ $$T^{2} + 338724$$
$59$ $$T^{2} + 556516$$
$61$ $$T^{2} + 51076$$
$67$ $$T^{2} + 29584$$
$71$ $$(T + 928)^{2}$$
$73$ $$(T + 570)^{2}$$
$79$ $$(T - 64)^{2}$$
$83$ $$T^{2} + 746496$$
$89$ $$(T - 874)^{2}$$
$97$ $$(T - 306)^{2}$$