# Properties

 Label 1280.3.e.c Level $1280$ Weight $3$ Character orbit 1280.e Analytic conductor $34.877$ Analytic rank $0$ Dimension $2$ CM discriminant -20 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1280,3,Mod(639,1280)]

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([1, 1, 1]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1280.639");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1280 = 2^{8} \cdot 5$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1280.e (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: $$34.8774738381$$ 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: $\mathrm{U}(1)[D_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + 4 i q^{3} + 5 i q^{5} + 4 q^{7} - 7 q^{9}+O(q^{10})$$ q + 4*i * q^3 + 5*i * q^5 + 4 * q^7 - 7 * q^9 $$q + 4 i q^{3} + 5 i q^{5} + 4 q^{7} - 7 q^{9} - 20 q^{15} + 16 i q^{21} - 44 q^{23} - 25 q^{25} + 8 i q^{27} - 22 i q^{29} + 20 i q^{35} - 62 q^{41} + 76 i q^{43} - 35 i q^{45} - 4 q^{47} - 33 q^{49} - 58 i q^{61} - 28 q^{63} + 116 i q^{67} - 176 i q^{69} - 100 i q^{75} - 95 q^{81} - 76 i q^{83} + 88 q^{87} + 142 q^{89} +O(q^{100})$$ q + 4*i * q^3 + 5*i * q^5 + 4 * q^7 - 7 * q^9 - 20 * q^15 + 16*i * q^21 - 44 * q^23 - 25 * q^25 + 8*i * q^27 - 22*i * q^29 + 20*i * q^35 - 62 * q^41 + 76*i * q^43 - 35*i * q^45 - 4 * q^47 - 33 * q^49 - 58*i * q^61 - 28 * q^63 + 116*i * q^67 - 176*i * q^69 - 100*i * q^75 - 95 * q^81 - 76*i * q^83 + 88 * q^87 + 142 * q^89 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 8 q^{7} - 14 q^{9}+O(q^{10})$$ 2 * q + 8 * q^7 - 14 * q^9 $$2 q + 8 q^{7} - 14 q^{9} - 40 q^{15} - 88 q^{23} - 50 q^{25} - 124 q^{41} - 8 q^{47} - 66 q^{49} - 56 q^{63} - 190 q^{81} + 176 q^{87} + 284 q^{89}+O(q^{100})$$ 2 * q + 8 * q^7 - 14 * q^9 - 40 * q^15 - 88 * q^23 - 50 * q^25 - 124 * q^41 - 8 * q^47 - 66 * q^49 - 56 * q^63 - 190 * q^81 + 176 * q^87 + 284 * q^89

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
639.1
 − 1.00000i 1.00000i
0 4.00000i 0 5.00000i 0 4.00000 0 −7.00000 0
639.2 0 4.00000i 0 5.00000i 0 4.00000 0 −7.00000 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
20.d odd 2 1 CM by $$\Q(\sqrt{-5})$$
8.b even 2 1 inner
40.e odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1280.3.e.c 2
4.b odd 2 1 1280.3.e.b 2
5.b even 2 1 1280.3.e.b 2
8.b even 2 1 inner 1280.3.e.c 2
8.d odd 2 1 1280.3.e.b 2
16.e even 4 1 20.3.d.a 1
16.e even 4 1 320.3.h.a 1
16.f odd 4 1 20.3.d.b yes 1
16.f odd 4 1 320.3.h.b 1
20.d odd 2 1 CM 1280.3.e.c 2
40.e odd 2 1 inner 1280.3.e.c 2
40.f even 2 1 1280.3.e.b 2
48.i odd 4 1 180.3.f.b 1
48.k even 4 1 180.3.f.a 1
80.i odd 4 1 100.3.b.c 2
80.i odd 4 1 1600.3.b.f 2
80.j even 4 1 100.3.b.c 2
80.j even 4 1 1600.3.b.f 2
80.k odd 4 1 20.3.d.a 1
80.k odd 4 1 320.3.h.a 1
80.q even 4 1 20.3.d.b yes 1
80.q even 4 1 320.3.h.b 1
80.s even 4 1 100.3.b.c 2
80.s even 4 1 1600.3.b.f 2
80.t odd 4 1 100.3.b.c 2
80.t odd 4 1 1600.3.b.f 2
240.t even 4 1 180.3.f.b 1
240.z odd 4 1 900.3.c.h 2
240.bb even 4 1 900.3.c.h 2
240.bd odd 4 1 900.3.c.h 2
240.bf even 4 1 900.3.c.h 2
240.bm odd 4 1 180.3.f.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.3.d.a 1 16.e even 4 1
20.3.d.a 1 80.k odd 4 1
20.3.d.b yes 1 16.f odd 4 1
20.3.d.b yes 1 80.q even 4 1
100.3.b.c 2 80.i odd 4 1
100.3.b.c 2 80.j even 4 1
100.3.b.c 2 80.s even 4 1
100.3.b.c 2 80.t odd 4 1
180.3.f.a 1 48.k even 4 1
180.3.f.a 1 240.bm odd 4 1
180.3.f.b 1 48.i odd 4 1
180.3.f.b 1 240.t even 4 1
320.3.h.a 1 16.e even 4 1
320.3.h.a 1 80.k odd 4 1
320.3.h.b 1 16.f odd 4 1
320.3.h.b 1 80.q even 4 1
900.3.c.h 2 240.z odd 4 1
900.3.c.h 2 240.bb even 4 1
900.3.c.h 2 240.bd odd 4 1
900.3.c.h 2 240.bf even 4 1
1280.3.e.b 2 4.b odd 2 1
1280.3.e.b 2 5.b even 2 1
1280.3.e.b 2 8.d odd 2 1
1280.3.e.b 2 40.f even 2 1
1280.3.e.c 2 1.a even 1 1 trivial
1280.3.e.c 2 8.b even 2 1 inner
1280.3.e.c 2 20.d odd 2 1 CM
1280.3.e.c 2 40.e odd 2 1 inner
1600.3.b.f 2 80.i odd 4 1
1600.3.b.f 2 80.j even 4 1
1600.3.b.f 2 80.s even 4 1
1600.3.b.f 2 80.t odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(1280, [\chi])$$:

 $$T_{3}^{2} + 16$$ T3^2 + 16 $$T_{7} - 4$$ T7 - 4 $$T_{11}$$ T11 $$T_{13}$$ T13

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 16$$
$5$ $$T^{2} + 25$$
$7$ $$(T - 4)^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2}$$
$17$ $$T^{2}$$
$19$ $$T^{2}$$
$23$ $$(T + 44)^{2}$$
$29$ $$T^{2} + 484$$
$31$ $$T^{2}$$
$37$ $$T^{2}$$
$41$ $$(T + 62)^{2}$$
$43$ $$T^{2} + 5776$$
$47$ $$(T + 4)^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$T^{2} + 3364$$
$67$ $$T^{2} + 13456$$
$71$ $$T^{2}$$
$73$ $$T^{2}$$
$79$ $$T^{2}$$
$83$ $$T^{2} + 5776$$
$89$ $$(T - 142)^{2}$$
$97$ $$T^{2}$$