# Properties

 Label 1280.4.d.l 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 5) 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.

 $$f(q)$$ $$=$$ $$q + 2 i q^{3} - 5 i q^{5} + 6 q^{7} + 23 q^{9} +O(q^{10})$$ q + 2*i * q^3 - 5*i * q^5 + 6 * q^7 + 23 * q^9 $$q + 2 i q^{3} - 5 i q^{5} + 6 q^{7} + 23 q^{9} - 32 i q^{11} + 38 i q^{13} + 10 q^{15} + 26 q^{17} + 100 i q^{19} + 12 i q^{21} - 78 q^{23} - 25 q^{25} + 100 i q^{27} + 50 i q^{29} + 108 q^{31} + 64 q^{33} - 30 i q^{35} + 266 i q^{37} - 76 q^{39} - 22 q^{41} - 442 i q^{43} - 115 i q^{45} + 514 q^{47} - 307 q^{49} + 52 i q^{51} + 2 i q^{53} - 160 q^{55} - 200 q^{57} - 500 i q^{59} + 518 i q^{61} + 138 q^{63} + 190 q^{65} + 126 i q^{67} - 156 i q^{69} + 412 q^{71} + 878 q^{73} - 50 i q^{75} - 192 i q^{77} - 600 q^{79} + 421 q^{81} + 282 i q^{83} - 130 i q^{85} - 100 q^{87} + 150 q^{89} + 228 i q^{91} + 216 i q^{93} + 500 q^{95} + 386 q^{97} - 736 i q^{99} +O(q^{100})$$ q + 2*i * q^3 - 5*i * q^5 + 6 * q^7 + 23 * q^9 - 32*i * q^11 + 38*i * q^13 + 10 * q^15 + 26 * q^17 + 100*i * q^19 + 12*i * q^21 - 78 * q^23 - 25 * q^25 + 100*i * q^27 + 50*i * q^29 + 108 * q^31 + 64 * q^33 - 30*i * q^35 + 266*i * q^37 - 76 * q^39 - 22 * q^41 - 442*i * q^43 - 115*i * q^45 + 514 * q^47 - 307 * q^49 + 52*i * q^51 + 2*i * q^53 - 160 * q^55 - 200 * q^57 - 500*i * q^59 + 518*i * q^61 + 138 * q^63 + 190 * q^65 + 126*i * q^67 - 156*i * q^69 + 412 * q^71 + 878 * q^73 - 50*i * q^75 - 192*i * q^77 - 600 * q^79 + 421 * q^81 + 282*i * q^83 - 130*i * q^85 - 100 * q^87 + 150 * q^89 + 228*i * q^91 + 216*i * q^93 + 500 * q^95 + 386 * q^97 - 736*i * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 12 q^{7} + 46 q^{9}+O(q^{10})$$ 2 * q + 12 * q^7 + 46 * q^9 $$2 q + 12 q^{7} + 46 q^{9} + 20 q^{15} + 52 q^{17} - 156 q^{23} - 50 q^{25} + 216 q^{31} + 128 q^{33} - 152 q^{39} - 44 q^{41} + 1028 q^{47} - 614 q^{49} - 320 q^{55} - 400 q^{57} + 276 q^{63} + 380 q^{65} + 824 q^{71} + 1756 q^{73} - 1200 q^{79} + 842 q^{81} - 200 q^{87} + 300 q^{89} + 1000 q^{95} + 772 q^{97}+O(q^{100})$$ 2 * q + 12 * q^7 + 46 * q^9 + 20 * q^15 + 52 * q^17 - 156 * q^23 - 50 * q^25 + 216 * q^31 + 128 * q^33 - 152 * q^39 - 44 * q^41 + 1028 * q^47 - 614 * q^49 - 320 * q^55 - 400 * q^57 + 276 * q^63 + 380 * q^65 + 824 * q^71 + 1756 * q^73 - 1200 * q^79 + 842 * q^81 - 200 * q^87 + 300 * q^89 + 1000 * q^95 + 772 * 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 2.00000i 0 5.00000i 0 6.00000 0 23.0000 0
641.2 0 2.00000i 0 5.00000i 0 6.00000 0 23.0000 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.l 2
4.b odd 2 1 1280.4.d.e 2
8.b even 2 1 inner 1280.4.d.l 2
8.d odd 2 1 1280.4.d.e 2
16.e even 4 1 80.4.a.d 1
16.e even 4 1 320.4.a.h 1
16.f odd 4 1 5.4.a.a 1
16.f odd 4 1 320.4.a.g 1
48.i odd 4 1 720.4.a.u 1
48.k even 4 1 45.4.a.d 1
80.i odd 4 1 400.4.c.k 2
80.j even 4 1 25.4.b.a 2
80.k odd 4 1 25.4.a.c 1
80.k odd 4 1 1600.4.a.bi 1
80.q even 4 1 400.4.a.m 1
80.q even 4 1 1600.4.a.s 1
80.s even 4 1 25.4.b.a 2
80.t odd 4 1 400.4.c.k 2
112.j even 4 1 245.4.a.a 1
112.u odd 12 2 245.4.e.f 2
112.v even 12 2 245.4.e.g 2
144.u even 12 2 405.4.e.c 2
144.v odd 12 2 405.4.e.l 2
176.i even 4 1 605.4.a.d 1
208.o odd 4 1 845.4.a.b 1
240.t even 4 1 225.4.a.b 1
240.z odd 4 1 225.4.b.c 2
240.bd odd 4 1 225.4.b.c 2
272.k odd 4 1 1445.4.a.a 1
304.m even 4 1 1805.4.a.h 1
336.v odd 4 1 2205.4.a.q 1
560.be even 4 1 1225.4.a.k 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.4.a.a 1 16.f odd 4 1
25.4.a.c 1 80.k odd 4 1
25.4.b.a 2 80.j even 4 1
25.4.b.a 2 80.s even 4 1
45.4.a.d 1 48.k even 4 1
80.4.a.d 1 16.e even 4 1
225.4.a.b 1 240.t even 4 1
225.4.b.c 2 240.z odd 4 1
225.4.b.c 2 240.bd odd 4 1
245.4.a.a 1 112.j even 4 1
245.4.e.f 2 112.u odd 12 2
245.4.e.g 2 112.v even 12 2
320.4.a.g 1 16.f odd 4 1
320.4.a.h 1 16.e even 4 1
400.4.a.m 1 80.q even 4 1
400.4.c.k 2 80.i odd 4 1
400.4.c.k 2 80.t odd 4 1
405.4.e.c 2 144.u even 12 2
405.4.e.l 2 144.v odd 12 2
605.4.a.d 1 176.i even 4 1
720.4.a.u 1 48.i odd 4 1
845.4.a.b 1 208.o odd 4 1
1225.4.a.k 1 560.be even 4 1
1280.4.d.e 2 4.b odd 2 1
1280.4.d.e 2 8.d odd 2 1
1280.4.d.l 2 1.a even 1 1 trivial
1280.4.d.l 2 8.b even 2 1 inner
1445.4.a.a 1 272.k odd 4 1
1600.4.a.s 1 80.q even 4 1
1600.4.a.bi 1 80.k odd 4 1
1805.4.a.h 1 304.m even 4 1
2205.4.a.q 1 336.v odd 4 1

## Hecke kernels

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

 $$T_{3}^{2} + 4$$ T3^2 + 4 $$T_{7} - 6$$ T7 - 6 $$T_{11}^{2} + 1024$$ T11^2 + 1024

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 4$$
$5$ $$T^{2} + 25$$
$7$ $$(T - 6)^{2}$$
$11$ $$T^{2} + 1024$$
$13$ $$T^{2} + 1444$$
$17$ $$(T - 26)^{2}$$
$19$ $$T^{2} + 10000$$
$23$ $$(T + 78)^{2}$$
$29$ $$T^{2} + 2500$$
$31$ $$(T - 108)^{2}$$
$37$ $$T^{2} + 70756$$
$41$ $$(T + 22)^{2}$$
$43$ $$T^{2} + 195364$$
$47$ $$(T - 514)^{2}$$
$53$ $$T^{2} + 4$$
$59$ $$T^{2} + 250000$$
$61$ $$T^{2} + 268324$$
$67$ $$T^{2} + 15876$$
$71$ $$(T - 412)^{2}$$
$73$ $$(T - 878)^{2}$$
$79$ $$(T + 600)^{2}$$
$83$ $$T^{2} + 79524$$
$89$ $$(T - 150)^{2}$$
$97$ $$(T - 386)^{2}$$