# Properties

 Label 1280.4.d.p 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 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.

 $$f(q)$$ $$=$$ $$q + 6 i q^{3} - 5 i q^{5} + 34 q^{7} - 9 q^{9} +O(q^{10})$$ q + 6*i * q^3 - 5*i * q^5 + 34 * q^7 - 9 * q^9 $$q + 6 i q^{3} - 5 i q^{5} + 34 q^{7} - 9 q^{9} + 16 i q^{11} - 58 i q^{13} + 30 q^{15} - 70 q^{17} - 4 i q^{19} + 204 i q^{21} + 134 q^{23} - 25 q^{25} + 108 i q^{27} + 242 i q^{29} + 100 q^{31} - 96 q^{33} - 170 i q^{35} - 438 i q^{37} + 348 q^{39} + 138 q^{41} + 178 i q^{43} + 45 i q^{45} + 22 q^{47} + 813 q^{49} - 420 i q^{51} + 162 i q^{53} + 80 q^{55} + 24 q^{57} - 268 i q^{59} - 250 i q^{61} - 306 q^{63} - 290 q^{65} - 422 i q^{67} + 804 i q^{69} + 852 q^{71} - 306 q^{73} - 150 i q^{75} + 544 i q^{77} - 456 q^{79} - 891 q^{81} - 434 i q^{83} + 350 i q^{85} - 1452 q^{87} + 726 q^{89} - 1972 i q^{91} + 600 i q^{93} - 20 q^{95} + 1378 q^{97} - 144 i q^{99} +O(q^{100})$$ q + 6*i * q^3 - 5*i * q^5 + 34 * q^7 - 9 * q^9 + 16*i * q^11 - 58*i * q^13 + 30 * q^15 - 70 * q^17 - 4*i * q^19 + 204*i * q^21 + 134 * q^23 - 25 * q^25 + 108*i * q^27 + 242*i * q^29 + 100 * q^31 - 96 * q^33 - 170*i * q^35 - 438*i * q^37 + 348 * q^39 + 138 * q^41 + 178*i * q^43 + 45*i * q^45 + 22 * q^47 + 813 * q^49 - 420*i * q^51 + 162*i * q^53 + 80 * q^55 + 24 * q^57 - 268*i * q^59 - 250*i * q^61 - 306 * q^63 - 290 * q^65 - 422*i * q^67 + 804*i * q^69 + 852 * q^71 - 306 * q^73 - 150*i * q^75 + 544*i * q^77 - 456 * q^79 - 891 * q^81 - 434*i * q^83 + 350*i * q^85 - 1452 * q^87 + 726 * q^89 - 1972*i * q^91 + 600*i * q^93 - 20 * q^95 + 1378 * q^97 - 144*i * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 68 q^{7} - 18 q^{9}+O(q^{10})$$ 2 * q + 68 * q^7 - 18 * q^9 $$2 q + 68 q^{7} - 18 q^{9} + 60 q^{15} - 140 q^{17} + 268 q^{23} - 50 q^{25} + 200 q^{31} - 192 q^{33} + 696 q^{39} + 276 q^{41} + 44 q^{47} + 1626 q^{49} + 160 q^{55} + 48 q^{57} - 612 q^{63} - 580 q^{65} + 1704 q^{71} - 612 q^{73} - 912 q^{79} - 1782 q^{81} - 2904 q^{87} + 1452 q^{89} - 40 q^{95} + 2756 q^{97}+O(q^{100})$$ 2 * q + 68 * q^7 - 18 * q^9 + 60 * q^15 - 140 * q^17 + 268 * q^23 - 50 * q^25 + 200 * q^31 - 192 * q^33 + 696 * q^39 + 276 * q^41 + 44 * q^47 + 1626 * q^49 + 160 * q^55 + 48 * q^57 - 612 * q^63 - 580 * q^65 + 1704 * q^71 - 612 * q^73 - 912 * q^79 - 1782 * q^81 - 2904 * q^87 + 1452 * q^89 - 40 * q^95 + 2756 * 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 6.00000i 0 5.00000i 0 34.0000 0 −9.00000 0
641.2 0 6.00000i 0 5.00000i 0 34.0000 0 −9.00000 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.p 2
4.b odd 2 1 1280.4.d.a 2
8.b even 2 1 inner 1280.4.d.p 2
8.d odd 2 1 1280.4.d.a 2
16.e even 4 1 40.4.a.a 1
16.e even 4 1 320.4.a.l 1
16.f odd 4 1 80.4.a.e 1
16.f odd 4 1 320.4.a.c 1
48.i odd 4 1 360.4.a.h 1
48.k even 4 1 720.4.a.bd 1
80.i odd 4 1 200.4.c.c 2
80.j even 4 1 400.4.c.f 2
80.k odd 4 1 400.4.a.e 1
80.k odd 4 1 1600.4.a.br 1
80.q even 4 1 200.4.a.i 1
80.q even 4 1 1600.4.a.j 1
80.s even 4 1 400.4.c.f 2
80.t odd 4 1 200.4.c.c 2
112.l odd 4 1 1960.4.a.h 1
240.bb even 4 1 1800.4.f.j 2
240.bf even 4 1 1800.4.f.j 2
240.bm odd 4 1 1800.4.a.bi 1

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

 $$T_{3}^{2} + 36$$ T3^2 + 36 $$T_{7} - 34$$ T7 - 34 $$T_{11}^{2} + 256$$ T11^2 + 256

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 36$$
$5$ $$T^{2} + 25$$
$7$ $$(T - 34)^{2}$$
$11$ $$T^{2} + 256$$
$13$ $$T^{2} + 3364$$
$17$ $$(T + 70)^{2}$$
$19$ $$T^{2} + 16$$
$23$ $$(T - 134)^{2}$$
$29$ $$T^{2} + 58564$$
$31$ $$(T - 100)^{2}$$
$37$ $$T^{2} + 191844$$
$41$ $$(T - 138)^{2}$$
$43$ $$T^{2} + 31684$$
$47$ $$(T - 22)^{2}$$
$53$ $$T^{2} + 26244$$
$59$ $$T^{2} + 71824$$
$61$ $$T^{2} + 62500$$
$67$ $$T^{2} + 178084$$
$71$ $$(T - 852)^{2}$$
$73$ $$(T + 306)^{2}$$
$79$ $$(T + 456)^{2}$$
$83$ $$T^{2} + 188356$$
$89$ $$(T - 726)^{2}$$
$97$ $$(T - 1378)^{2}$$