# Properties

 Label 1280.4.d.o 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: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 40) 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.

 $$f(q)$$ $$=$$ $$q + 10 i q^{3} + 5 i q^{5} + 18 q^{7} - 73 q^{9}+O(q^{10})$$ q + 10*i * q^3 + 5*i * q^5 + 18 * q^7 - 73 * q^9 $$q + 10 i q^{3} + 5 i q^{5} + 18 q^{7} - 73 q^{9} + 16 i q^{11} - 6 i q^{13} - 50 q^{15} - 6 q^{17} - 124 i q^{19} + 180 i q^{21} - 42 q^{23} - 25 q^{25} - 460 i q^{27} + 142 i q^{29} - 188 q^{31} - 160 q^{33} + 90 i q^{35} - 202 i q^{37} + 60 q^{39} - 54 q^{41} - 66 i q^{43} - 365 i q^{45} + 38 q^{47} - 19 q^{49} - 60 i q^{51} - 738 i q^{53} - 80 q^{55} + 1240 q^{57} - 564 i q^{59} - 262 i q^{61} - 1314 q^{63} + 30 q^{65} - 554 i q^{67} - 420 i q^{69} - 140 q^{71} - 882 q^{73} - 250 i q^{75} + 288 i q^{77} - 1160 q^{79} + 2629 q^{81} + 642 i q^{83} - 30 i q^{85} - 1420 q^{87} + 854 q^{89} - 108 i q^{91} - 1880 i q^{93} + 620 q^{95} - 478 q^{97} - 1168 i q^{99} +O(q^{100})$$ q + 10*i * q^3 + 5*i * q^5 + 18 * q^7 - 73 * q^9 + 16*i * q^11 - 6*i * q^13 - 50 * q^15 - 6 * q^17 - 124*i * q^19 + 180*i * q^21 - 42 * q^23 - 25 * q^25 - 460*i * q^27 + 142*i * q^29 - 188 * q^31 - 160 * q^33 + 90*i * q^35 - 202*i * q^37 + 60 * q^39 - 54 * q^41 - 66*i * q^43 - 365*i * q^45 + 38 * q^47 - 19 * q^49 - 60*i * q^51 - 738*i * q^53 - 80 * q^55 + 1240 * q^57 - 564*i * q^59 - 262*i * q^61 - 1314 * q^63 + 30 * q^65 - 554*i * q^67 - 420*i * q^69 - 140 * q^71 - 882 * q^73 - 250*i * q^75 + 288*i * q^77 - 1160 * q^79 + 2629 * q^81 + 642*i * q^83 - 30*i * q^85 - 1420 * q^87 + 854 * q^89 - 108*i * q^91 - 1880*i * q^93 + 620 * q^95 - 478 * q^97 - 1168*i * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 36 q^{7} - 146 q^{9}+O(q^{10})$$ 2 * q + 36 * q^7 - 146 * q^9 $$2 q + 36 q^{7} - 146 q^{9} - 100 q^{15} - 12 q^{17} - 84 q^{23} - 50 q^{25} - 376 q^{31} - 320 q^{33} + 120 q^{39} - 108 q^{41} + 76 q^{47} - 38 q^{49} - 160 q^{55} + 2480 q^{57} - 2628 q^{63} + 60 q^{65} - 280 q^{71} - 1764 q^{73} - 2320 q^{79} + 5258 q^{81} - 2840 q^{87} + 1708 q^{89} + 1240 q^{95} - 956 q^{97}+O(q^{100})$$ 2 * q + 36 * q^7 - 146 * q^9 - 100 * q^15 - 12 * q^17 - 84 * q^23 - 50 * q^25 - 376 * q^31 - 320 * q^33 + 120 * q^39 - 108 * q^41 + 76 * q^47 - 38 * q^49 - 160 * q^55 + 2480 * q^57 - 2628 * q^63 + 60 * q^65 - 280 * q^71 - 1764 * q^73 - 2320 * q^79 + 5258 * q^81 - 2840 * q^87 + 1708 * q^89 + 1240 * q^95 - 956 * 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 10.0000i 0 5.00000i 0 18.0000 0 −73.0000 0
641.2 0 10.0000i 0 5.00000i 0 18.0000 0 −73.0000 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.o 2
4.b odd 2 1 1280.4.d.b 2
8.b even 2 1 inner 1280.4.d.o 2
8.d odd 2 1 1280.4.d.b 2
16.e even 4 1 40.4.a.c 1
16.e even 4 1 320.4.a.a 1
16.f odd 4 1 80.4.a.a 1
16.f odd 4 1 320.4.a.n 1
48.i odd 4 1 360.4.a.i 1
48.k even 4 1 720.4.a.ba 1
80.i odd 4 1 200.4.c.a 2
80.j even 4 1 400.4.c.a 2
80.k odd 4 1 400.4.a.u 1
80.k odd 4 1 1600.4.a.a 1
80.q even 4 1 200.4.a.a 1
80.q even 4 1 1600.4.a.ca 1
80.s even 4 1 400.4.c.a 2
80.t odd 4 1 200.4.c.a 2
112.l odd 4 1 1960.4.a.a 1
240.bb even 4 1 1800.4.f.n 2
240.bf even 4 1 1800.4.f.n 2
240.bm odd 4 1 1800.4.a.bd 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.4.a.c 1 16.e even 4 1
80.4.a.a 1 16.f odd 4 1
200.4.a.a 1 80.q even 4 1
200.4.c.a 2 80.i odd 4 1
200.4.c.a 2 80.t odd 4 1
320.4.a.a 1 16.e even 4 1
320.4.a.n 1 16.f odd 4 1
360.4.a.i 1 48.i odd 4 1
400.4.a.u 1 80.k odd 4 1
400.4.c.a 2 80.j even 4 1
400.4.c.a 2 80.s even 4 1
720.4.a.ba 1 48.k even 4 1
1280.4.d.b 2 4.b odd 2 1
1280.4.d.b 2 8.d odd 2 1
1280.4.d.o 2 1.a even 1 1 trivial
1280.4.d.o 2 8.b even 2 1 inner
1600.4.a.a 1 80.k odd 4 1
1600.4.a.ca 1 80.q even 4 1
1800.4.a.bd 1 240.bm odd 4 1
1800.4.f.n 2 240.bb even 4 1
1800.4.f.n 2 240.bf even 4 1
1960.4.a.a 1 112.l odd 4 1

## Hecke kernels

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

 $$T_{3}^{2} + 100$$ T3^2 + 100 $$T_{7} - 18$$ T7 - 18 $$T_{11}^{2} + 256$$ T11^2 + 256

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 100$$
$5$ $$T^{2} + 25$$
$7$ $$(T - 18)^{2}$$
$11$ $$T^{2} + 256$$
$13$ $$T^{2} + 36$$
$17$ $$(T + 6)^{2}$$
$19$ $$T^{2} + 15376$$
$23$ $$(T + 42)^{2}$$
$29$ $$T^{2} + 20164$$
$31$ $$(T + 188)^{2}$$
$37$ $$T^{2} + 40804$$
$41$ $$(T + 54)^{2}$$
$43$ $$T^{2} + 4356$$
$47$ $$(T - 38)^{2}$$
$53$ $$T^{2} + 544644$$
$59$ $$T^{2} + 318096$$
$61$ $$T^{2} + 68644$$
$67$ $$T^{2} + 306916$$
$71$ $$(T + 140)^{2}$$
$73$ $$(T + 882)^{2}$$
$79$ $$(T + 1160)^{2}$$
$83$ $$T^{2} + 412164$$
$89$ $$(T - 854)^{2}$$
$97$ $$(T + 478)^{2}$$