# Properties

 Label 1280.4.d.c 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 20) 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.

 $$f(q)$$ $$=$$ $$q + 4 i q^{3} + 5 i q^{5} - 16 q^{7} + 11 q^{9}+O(q^{10})$$ q + 4*i * q^3 + 5*i * q^5 - 16 * q^7 + 11 * q^9 $$q + 4 i q^{3} + 5 i q^{5} - 16 q^{7} + 11 q^{9} + 60 i q^{11} - 86 i q^{13} - 20 q^{15} + 18 q^{17} + 44 i q^{19} - 64 i q^{21} + 48 q^{23} - 25 q^{25} + 152 i q^{27} + 186 i q^{29} - 176 q^{31} - 240 q^{33} - 80 i q^{35} + 254 i q^{37} + 344 q^{39} - 186 q^{41} + 100 i q^{43} + 55 i q^{45} - 168 q^{47} - 87 q^{49} + 72 i q^{51} - 498 i q^{53} - 300 q^{55} - 176 q^{57} + 252 i q^{59} + 58 i q^{61} - 176 q^{63} + 430 q^{65} - 1036 i q^{67} + 192 i q^{69} + 168 q^{71} - 506 q^{73} - 100 i q^{75} - 960 i q^{77} - 272 q^{79} - 311 q^{81} + 948 i q^{83} + 90 i q^{85} - 744 q^{87} + 1014 q^{89} + 1376 i q^{91} - 704 i q^{93} - 220 q^{95} - 766 q^{97} + 660 i q^{99} +O(q^{100})$$ q + 4*i * q^3 + 5*i * q^5 - 16 * q^7 + 11 * q^9 + 60*i * q^11 - 86*i * q^13 - 20 * q^15 + 18 * q^17 + 44*i * q^19 - 64*i * q^21 + 48 * q^23 - 25 * q^25 + 152*i * q^27 + 186*i * q^29 - 176 * q^31 - 240 * q^33 - 80*i * q^35 + 254*i * q^37 + 344 * q^39 - 186 * q^41 + 100*i * q^43 + 55*i * q^45 - 168 * q^47 - 87 * q^49 + 72*i * q^51 - 498*i * q^53 - 300 * q^55 - 176 * q^57 + 252*i * q^59 + 58*i * q^61 - 176 * q^63 + 430 * q^65 - 1036*i * q^67 + 192*i * q^69 + 168 * q^71 - 506 * q^73 - 100*i * q^75 - 960*i * q^77 - 272 * q^79 - 311 * q^81 + 948*i * q^83 + 90*i * q^85 - 744 * q^87 + 1014 * q^89 + 1376*i * q^91 - 704*i * q^93 - 220 * q^95 - 766 * q^97 + 660*i * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 32 q^{7} + 22 q^{9}+O(q^{10})$$ 2 * q - 32 * q^7 + 22 * q^9 $$2 q - 32 q^{7} + 22 q^{9} - 40 q^{15} + 36 q^{17} + 96 q^{23} - 50 q^{25} - 352 q^{31} - 480 q^{33} + 688 q^{39} - 372 q^{41} - 336 q^{47} - 174 q^{49} - 600 q^{55} - 352 q^{57} - 352 q^{63} + 860 q^{65} + 336 q^{71} - 1012 q^{73} - 544 q^{79} - 622 q^{81} - 1488 q^{87} + 2028 q^{89} - 440 q^{95} - 1532 q^{97}+O(q^{100})$$ 2 * q - 32 * q^7 + 22 * q^9 - 40 * q^15 + 36 * q^17 + 96 * q^23 - 50 * q^25 - 352 * q^31 - 480 * q^33 + 688 * q^39 - 372 * q^41 - 336 * q^47 - 174 * q^49 - 600 * q^55 - 352 * q^57 - 352 * q^63 + 860 * q^65 + 336 * q^71 - 1012 * q^73 - 544 * q^79 - 622 * q^81 - 1488 * q^87 + 2028 * q^89 - 440 * q^95 - 1532 * 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 4.00000i 0 5.00000i 0 −16.0000 0 11.0000 0
641.2 0 4.00000i 0 5.00000i 0 −16.0000 0 11.0000 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.c 2
4.b odd 2 1 1280.4.d.n 2
8.b even 2 1 inner 1280.4.d.c 2
8.d odd 2 1 1280.4.d.n 2
16.e even 4 1 80.4.a.c 1
16.e even 4 1 320.4.a.k 1
16.f odd 4 1 20.4.a.a 1
16.f odd 4 1 320.4.a.d 1
48.i odd 4 1 720.4.a.k 1
48.k even 4 1 180.4.a.a 1
80.i odd 4 1 400.4.c.j 2
80.j even 4 1 100.4.c.a 2
80.k odd 4 1 100.4.a.a 1
80.k odd 4 1 1600.4.a.bl 1
80.q even 4 1 400.4.a.o 1
80.q even 4 1 1600.4.a.p 1
80.s even 4 1 100.4.c.a 2
80.t odd 4 1 400.4.c.j 2
112.j even 4 1 980.4.a.c 1
112.u odd 12 2 980.4.i.e 2
112.v even 12 2 980.4.i.n 2
144.u even 12 2 1620.4.i.j 2
144.v odd 12 2 1620.4.i.d 2
176.i even 4 1 2420.4.a.d 1
240.t even 4 1 900.4.a.m 1
240.z odd 4 1 900.4.d.k 2
240.bd odd 4 1 900.4.d.k 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.4.a.a 1 16.f odd 4 1
80.4.a.c 1 16.e even 4 1
100.4.a.a 1 80.k odd 4 1
100.4.c.a 2 80.j even 4 1
100.4.c.a 2 80.s even 4 1
180.4.a.a 1 48.k even 4 1
320.4.a.d 1 16.f odd 4 1
320.4.a.k 1 16.e even 4 1
400.4.a.o 1 80.q even 4 1
400.4.c.j 2 80.i odd 4 1
400.4.c.j 2 80.t odd 4 1
720.4.a.k 1 48.i odd 4 1
900.4.a.m 1 240.t even 4 1
900.4.d.k 2 240.z odd 4 1
900.4.d.k 2 240.bd odd 4 1
980.4.a.c 1 112.j even 4 1
980.4.i.e 2 112.u odd 12 2
980.4.i.n 2 112.v even 12 2
1280.4.d.c 2 1.a even 1 1 trivial
1280.4.d.c 2 8.b even 2 1 inner
1280.4.d.n 2 4.b odd 2 1
1280.4.d.n 2 8.d odd 2 1
1600.4.a.p 1 80.q even 4 1
1600.4.a.bl 1 80.k odd 4 1
1620.4.i.d 2 144.v odd 12 2
1620.4.i.j 2 144.u even 12 2
2420.4.a.d 1 176.i even 4 1

## Hecke kernels

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

 $$T_{3}^{2} + 16$$ T3^2 + 16 $$T_{7} + 16$$ T7 + 16 $$T_{11}^{2} + 3600$$ T11^2 + 3600

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 16$$
$5$ $$T^{2} + 25$$
$7$ $$(T + 16)^{2}$$
$11$ $$T^{2} + 3600$$
$13$ $$T^{2} + 7396$$
$17$ $$(T - 18)^{2}$$
$19$ $$T^{2} + 1936$$
$23$ $$(T - 48)^{2}$$
$29$ $$T^{2} + 34596$$
$31$ $$(T + 176)^{2}$$
$37$ $$T^{2} + 64516$$
$41$ $$(T + 186)^{2}$$
$43$ $$T^{2} + 10000$$
$47$ $$(T + 168)^{2}$$
$53$ $$T^{2} + 248004$$
$59$ $$T^{2} + 63504$$
$61$ $$T^{2} + 3364$$
$67$ $$T^{2} + 1073296$$
$71$ $$(T - 168)^{2}$$
$73$ $$(T + 506)^{2}$$
$79$ $$(T + 272)^{2}$$
$83$ $$T^{2} + 898704$$
$89$ $$(T - 1014)^{2}$$
$97$ $$(T + 766)^{2}$$