# Properties

 Label 1280.4.d.d Level $1280$ Weight $4$ Character orbit 1280.d Analytic conductor $75.522$ Analytic rank $0$ Dimension $2$ 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 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.

 $$f(q)$$ $$=$$ $$q + 4 i q^{3} - 5 i q^{5} - 16 q^{7} + 11 q^{9} +O(q^{10})$$ q + 4*i * q^3 - 5*i * q^5 - 16 * q^7 + 11 * q^9 $$q + 4 i q^{3} - 5 i q^{5} - 16 q^{7} + 11 q^{9} - 36 i q^{11} - 42 i q^{13} + 20 q^{15} - 110 q^{17} - 116 i q^{19} - 64 i q^{21} - 16 q^{23} - 25 q^{25} + 152 i q^{27} + 198 i q^{29} + 240 q^{31} + 144 q^{33} + 80 i q^{35} + 258 i q^{37} + 168 q^{39} - 442 q^{41} + 292 i q^{43} - 55 i q^{45} + 392 q^{47} - 87 q^{49} - 440 i q^{51} - 142 i q^{53} - 180 q^{55} + 464 q^{57} + 348 i q^{59} - 570 i q^{61} - 176 q^{63} - 210 q^{65} + 692 i q^{67} - 64 i q^{69} - 168 q^{71} + 134 q^{73} - 100 i q^{75} + 576 i q^{77} + 784 q^{79} - 311 q^{81} + 564 i q^{83} + 550 i q^{85} - 792 q^{87} - 1034 q^{89} + 672 i q^{91} + 960 i q^{93} - 580 q^{95} - 382 q^{97} - 396 i q^{99} +O(q^{100})$$ q + 4*i * q^3 - 5*i * q^5 - 16 * q^7 + 11 * q^9 - 36*i * q^11 - 42*i * q^13 + 20 * q^15 - 110 * q^17 - 116*i * q^19 - 64*i * q^21 - 16 * q^23 - 25 * q^25 + 152*i * q^27 + 198*i * q^29 + 240 * q^31 + 144 * q^33 + 80*i * q^35 + 258*i * q^37 + 168 * q^39 - 442 * q^41 + 292*i * q^43 - 55*i * q^45 + 392 * q^47 - 87 * q^49 - 440*i * q^51 - 142*i * q^53 - 180 * q^55 + 464 * q^57 + 348*i * q^59 - 570*i * q^61 - 176 * q^63 - 210 * q^65 + 692*i * q^67 - 64*i * q^69 - 168 * q^71 + 134 * q^73 - 100*i * q^75 + 576*i * q^77 + 784 * q^79 - 311 * q^81 + 564*i * q^83 + 550*i * q^85 - 792 * q^87 - 1034 * q^89 + 672*i * q^91 + 960*i * q^93 - 580 * q^95 - 382 * q^97 - 396*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 $$2 q - 32 q^{7} + 22 q^{9} + 40 q^{15} - 220 q^{17} - 32 q^{23} - 50 q^{25} + 480 q^{31} + 288 q^{33} + 336 q^{39} - 884 q^{41} + 784 q^{47} - 174 q^{49} - 360 q^{55} + 928 q^{57} - 352 q^{63} - 420 q^{65} - 336 q^{71} + 268 q^{73} + 1568 q^{79} - 622 q^{81} - 1584 q^{87} - 2068 q^{89} - 1160 q^{95} - 764 q^{97}+O(q^{100})$$ 2 * q - 32 * q^7 + 22 * q^9 + 40 * q^15 - 220 * q^17 - 32 * q^23 - 50 * q^25 + 480 * q^31 + 288 * q^33 + 336 * q^39 - 884 * q^41 + 784 * q^47 - 174 * q^49 - 360 * q^55 + 928 * q^57 - 352 * q^63 - 420 * q^65 - 336 * q^71 + 268 * q^73 + 1568 * q^79 - 622 * q^81 - 1584 * q^87 - 2068 * q^89 - 1160 * q^95 - 764 * 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.d 2
4.b odd 2 1 1280.4.d.m 2
8.b even 2 1 inner 1280.4.d.d 2
8.d odd 2 1 1280.4.d.m 2
16.e even 4 1 40.4.a.b 1
16.e even 4 1 320.4.a.e 1
16.f odd 4 1 80.4.a.b 1
16.f odd 4 1 320.4.a.j 1
48.i odd 4 1 360.4.a.f 1
48.k even 4 1 720.4.a.d 1
80.i odd 4 1 200.4.c.f 2
80.j even 4 1 400.4.c.h 2
80.k odd 4 1 400.4.a.p 1
80.k odd 4 1 1600.4.a.q 1
80.q even 4 1 200.4.a.d 1
80.q even 4 1 1600.4.a.bk 1
80.s even 4 1 400.4.c.h 2
80.t odd 4 1 200.4.c.f 2
112.l odd 4 1 1960.4.a.e 1
240.bb even 4 1 1800.4.f.d 2
240.bf even 4 1 1800.4.f.d 2
240.bm odd 4 1 1800.4.a.h 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.4.a.b 1 16.e even 4 1
80.4.a.b 1 16.f odd 4 1
200.4.a.d 1 80.q even 4 1
200.4.c.f 2 80.i odd 4 1
200.4.c.f 2 80.t odd 4 1
320.4.a.e 1 16.e even 4 1
320.4.a.j 1 16.f odd 4 1
360.4.a.f 1 48.i odd 4 1
400.4.a.p 1 80.k odd 4 1
400.4.c.h 2 80.j even 4 1
400.4.c.h 2 80.s even 4 1
720.4.a.d 1 48.k even 4 1
1280.4.d.d 2 1.a even 1 1 trivial
1280.4.d.d 2 8.b even 2 1 inner
1280.4.d.m 2 4.b odd 2 1
1280.4.d.m 2 8.d odd 2 1
1600.4.a.q 1 80.k odd 4 1
1600.4.a.bk 1 80.q even 4 1
1800.4.a.h 1 240.bm odd 4 1
1800.4.f.d 2 240.bb even 4 1
1800.4.f.d 2 240.bf even 4 1
1960.4.a.e 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 $$S_{4}^{\mathrm{new}}(1280, [\chi])$$:

 $$T_{3}^{2} + 16$$ T3^2 + 16 $$T_{7} + 16$$ T7 + 16 $$T_{11}^{2} + 1296$$ T11^2 + 1296

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 16$$
$5$ $$T^{2} + 25$$
$7$ $$(T + 16)^{2}$$
$11$ $$T^{2} + 1296$$
$13$ $$T^{2} + 1764$$
$17$ $$(T + 110)^{2}$$
$19$ $$T^{2} + 13456$$
$23$ $$(T + 16)^{2}$$
$29$ $$T^{2} + 39204$$
$31$ $$(T - 240)^{2}$$
$37$ $$T^{2} + 66564$$
$41$ $$(T + 442)^{2}$$
$43$ $$T^{2} + 85264$$
$47$ $$(T - 392)^{2}$$
$53$ $$T^{2} + 20164$$
$59$ $$T^{2} + 121104$$
$61$ $$T^{2} + 324900$$
$67$ $$T^{2} + 478864$$
$71$ $$(T + 168)^{2}$$
$73$ $$(T - 134)^{2}$$
$79$ $$(T - 784)^{2}$$
$83$ $$T^{2} + 318096$$
$89$ $$(T + 1034)^{2}$$
$97$ $$(T + 382)^{2}$$