# Properties

 Label 1280.4.d.g 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 10) 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.

 $$f(q)$$ $$=$$ $$q + 8 i q^{3} - 5 i q^{5} - 4 q^{7} - 37 q^{9} +O(q^{10})$$ q + 8*i * q^3 - 5*i * q^5 - 4 * q^7 - 37 * q^9 $$q + 8 i q^{3} - 5 i q^{5} - 4 q^{7} - 37 q^{9} + 12 i q^{11} - 58 i q^{13} + 40 q^{15} + 66 q^{17} + 100 i q^{19} - 32 i q^{21} + 132 q^{23} - 25 q^{25} - 80 i q^{27} - 90 i q^{29} - 152 q^{31} - 96 q^{33} + 20 i q^{35} + 34 i q^{37} + 464 q^{39} + 438 q^{41} + 32 i q^{43} + 185 i q^{45} + 204 q^{47} - 327 q^{49} + 528 i q^{51} - 222 i q^{53} + 60 q^{55} - 800 q^{57} + 420 i q^{59} + 902 i q^{61} + 148 q^{63} - 290 q^{65} + 1024 i q^{67} + 1056 i q^{69} + 432 q^{71} - 362 q^{73} - 200 i q^{75} - 48 i q^{77} + 160 q^{79} - 359 q^{81} - 72 i q^{83} - 330 i q^{85} + 720 q^{87} - 810 q^{89} + 232 i q^{91} - 1216 i q^{93} + 500 q^{95} + 1106 q^{97} - 444 i q^{99} +O(q^{100})$$ q + 8*i * q^3 - 5*i * q^5 - 4 * q^7 - 37 * q^9 + 12*i * q^11 - 58*i * q^13 + 40 * q^15 + 66 * q^17 + 100*i * q^19 - 32*i * q^21 + 132 * q^23 - 25 * q^25 - 80*i * q^27 - 90*i * q^29 - 152 * q^31 - 96 * q^33 + 20*i * q^35 + 34*i * q^37 + 464 * q^39 + 438 * q^41 + 32*i * q^43 + 185*i * q^45 + 204 * q^47 - 327 * q^49 + 528*i * q^51 - 222*i * q^53 + 60 * q^55 - 800 * q^57 + 420*i * q^59 + 902*i * q^61 + 148 * q^63 - 290 * q^65 + 1024*i * q^67 + 1056*i * q^69 + 432 * q^71 - 362 * q^73 - 200*i * q^75 - 48*i * q^77 + 160 * q^79 - 359 * q^81 - 72*i * q^83 - 330*i * q^85 + 720 * q^87 - 810 * q^89 + 232*i * q^91 - 1216*i * q^93 + 500 * q^95 + 1106 * q^97 - 444*i * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 8 q^{7} - 74 q^{9}+O(q^{10})$$ 2 * q - 8 * q^7 - 74 * q^9 $$2 q - 8 q^{7} - 74 q^{9} + 80 q^{15} + 132 q^{17} + 264 q^{23} - 50 q^{25} - 304 q^{31} - 192 q^{33} + 928 q^{39} + 876 q^{41} + 408 q^{47} - 654 q^{49} + 120 q^{55} - 1600 q^{57} + 296 q^{63} - 580 q^{65} + 864 q^{71} - 724 q^{73} + 320 q^{79} - 718 q^{81} + 1440 q^{87} - 1620 q^{89} + 1000 q^{95} + 2212 q^{97}+O(q^{100})$$ 2 * q - 8 * q^7 - 74 * q^9 + 80 * q^15 + 132 * q^17 + 264 * q^23 - 50 * q^25 - 304 * q^31 - 192 * q^33 + 928 * q^39 + 876 * q^41 + 408 * q^47 - 654 * q^49 + 120 * q^55 - 1600 * q^57 + 296 * q^63 - 580 * q^65 + 864 * q^71 - 724 * q^73 + 320 * q^79 - 718 * q^81 + 1440 * q^87 - 1620 * q^89 + 1000 * q^95 + 2212 * 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 −4.00000 0 −37.0000 0
641.2 0 8.00000i 0 5.00000i 0 −4.00000 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.g 2
4.b odd 2 1 1280.4.d.j 2
8.b even 2 1 inner 1280.4.d.g 2
8.d odd 2 1 1280.4.d.j 2
16.e even 4 1 80.4.a.f 1
16.e even 4 1 320.4.a.b 1
16.f odd 4 1 10.4.a.a 1
16.f odd 4 1 320.4.a.m 1
48.i odd 4 1 720.4.a.j 1
48.k even 4 1 90.4.a.a 1
80.i odd 4 1 400.4.c.c 2
80.j even 4 1 50.4.b.a 2
80.k odd 4 1 50.4.a.c 1
80.k odd 4 1 1600.4.a.d 1
80.q even 4 1 400.4.a.b 1
80.q even 4 1 1600.4.a.bx 1
80.s even 4 1 50.4.b.a 2
80.t odd 4 1 400.4.c.c 2
112.j even 4 1 490.4.a.o 1
112.u odd 12 2 490.4.e.i 2
112.v even 12 2 490.4.e.a 2
144.u even 12 2 810.4.e.w 2
144.v odd 12 2 810.4.e.c 2
176.i even 4 1 1210.4.a.b 1
208.o odd 4 1 1690.4.a.a 1
240.t even 4 1 450.4.a.q 1
240.z odd 4 1 450.4.c.d 2
240.bd odd 4 1 450.4.c.d 2
560.be even 4 1 2450.4.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.4.a.a 1 16.f odd 4 1
50.4.a.c 1 80.k odd 4 1
50.4.b.a 2 80.j even 4 1
50.4.b.a 2 80.s even 4 1
80.4.a.f 1 16.e even 4 1
90.4.a.a 1 48.k even 4 1
320.4.a.b 1 16.e even 4 1
320.4.a.m 1 16.f odd 4 1
400.4.a.b 1 80.q even 4 1
400.4.c.c 2 80.i odd 4 1
400.4.c.c 2 80.t odd 4 1
450.4.a.q 1 240.t even 4 1
450.4.c.d 2 240.z odd 4 1
450.4.c.d 2 240.bd odd 4 1
490.4.a.o 1 112.j even 4 1
490.4.e.a 2 112.v even 12 2
490.4.e.i 2 112.u odd 12 2
720.4.a.j 1 48.i odd 4 1
810.4.e.c 2 144.v odd 12 2
810.4.e.w 2 144.u even 12 2
1210.4.a.b 1 176.i even 4 1
1280.4.d.g 2 1.a even 1 1 trivial
1280.4.d.g 2 8.b even 2 1 inner
1280.4.d.j 2 4.b odd 2 1
1280.4.d.j 2 8.d odd 2 1
1600.4.a.d 1 80.k odd 4 1
1600.4.a.bx 1 80.q even 4 1
1690.4.a.a 1 208.o odd 4 1
2450.4.a.b 1 560.be even 4 1

## Hecke kernels

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

 $$T_{3}^{2} + 64$$ T3^2 + 64 $$T_{7} + 4$$ T7 + 4 $$T_{11}^{2} + 144$$ T11^2 + 144

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 64$$
$5$ $$T^{2} + 25$$
$7$ $$(T + 4)^{2}$$
$11$ $$T^{2} + 144$$
$13$ $$T^{2} + 3364$$
$17$ $$(T - 66)^{2}$$
$19$ $$T^{2} + 10000$$
$23$ $$(T - 132)^{2}$$
$29$ $$T^{2} + 8100$$
$31$ $$(T + 152)^{2}$$
$37$ $$T^{2} + 1156$$
$41$ $$(T - 438)^{2}$$
$43$ $$T^{2} + 1024$$
$47$ $$(T - 204)^{2}$$
$53$ $$T^{2} + 49284$$
$59$ $$T^{2} + 176400$$
$61$ $$T^{2} + 813604$$
$67$ $$T^{2} + 1048576$$
$71$ $$(T - 432)^{2}$$
$73$ $$(T + 362)^{2}$$
$79$ $$(T - 160)^{2}$$
$83$ $$T^{2} + 5184$$
$89$ $$(T + 810)^{2}$$
$97$ $$(T - 1106)^{2}$$