# Properties

 Label 1280.2.o.r Level $1280$ Weight $2$ Character orbit 1280.o Analytic conductor $10.221$ Analytic rank $0$ Dimension $4$ Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1280,2,Mod(127,1280)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1280, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 2, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1280.127");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1280 = 2^{8} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1280.o (of order $$4$$, degree $$2$$, 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: $$10.2208514587$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 80) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{3} + (\beta_1 + 2) q^{5} - \beta_{2} q^{7} + 3 \beta_1 q^{9}+O(q^{10})$$ q - b2 * q^3 + (b1 + 2) * q^5 - b2 * q^7 + 3*b1 * q^9 $$q - \beta_{2} q^{3} + (\beta_1 + 2) q^{5} - \beta_{2} q^{7} + 3 \beta_1 q^{9} + ( - \beta_{3} - \beta_{2}) q^{11} + (\beta_1 - 1) q^{13} + (\beta_{3} - 2 \beta_{2}) q^{15} + ( - \beta_1 + 1) q^{17} + ( - 2 \beta_{3} + 2 \beta_{2}) q^{19} + 6 \beta_1 q^{21} + \beta_{3} q^{23} + (4 \beta_1 + 3) q^{25} - 4 q^{29} + ( - \beta_{3} + \beta_{2}) q^{31} + (6 \beta_1 + 6) q^{33} + (\beta_{3} - 2 \beta_{2}) q^{35} + ( - 5 \beta_1 - 5) q^{37} + (\beta_{3} + \beta_{2}) q^{39} - 2 q^{41} + \beta_{2} q^{43} + (6 \beta_1 - 3) q^{45} + \beta_{2} q^{47} - \beta_1 q^{49} + ( - \beta_{3} - \beta_{2}) q^{51} + (7 \beta_1 - 7) q^{53} + ( - \beta_{3} - 3 \beta_{2}) q^{55} + ( - 12 \beta_1 + 12) q^{57} + ( - 2 \beta_{3} + 2 \beta_{2}) q^{59} + 6 \beta_1 q^{61} + 3 \beta_{3} q^{63} + (\beta_1 - 3) q^{65} + 3 \beta_{3} q^{67} - 6 q^{69} + ( - 3 \beta_{3} + 3 \beta_{2}) q^{71} + (7 \beta_1 + 7) q^{73} + (4 \beta_{3} - 3 \beta_{2}) q^{75} + (6 \beta_1 + 6) q^{77} + 9 q^{81} + 7 \beta_{2} q^{83} + ( - \beta_1 + 3) q^{85} + 4 \beta_{2} q^{87} - 8 \beta_1 q^{89} + (\beta_{3} + \beta_{2}) q^{91} + ( - 6 \beta_1 + 6) q^{93} + ( - 6 \beta_{3} + 2 \beta_{2}) q^{95} + (7 \beta_1 - 7) q^{97} + (3 \beta_{3} - 3 \beta_{2}) q^{99}+O(q^{100})$$ q - b2 * q^3 + (b1 + 2) * q^5 - b2 * q^7 + 3*b1 * q^9 + (-b3 - b2) * q^11 + (b1 - 1) * q^13 + (b3 - 2*b2) * q^15 + (-b1 + 1) * q^17 + (-2*b3 + 2*b2) * q^19 + 6*b1 * q^21 + b3 * q^23 + (4*b1 + 3) * q^25 - 4 * q^29 + (-b3 + b2) * q^31 + (6*b1 + 6) * q^33 + (b3 - 2*b2) * q^35 + (-5*b1 - 5) * q^37 + (b3 + b2) * q^39 - 2 * q^41 + b2 * q^43 + (6*b1 - 3) * q^45 + b2 * q^47 - b1 * q^49 + (-b3 - b2) * q^51 + (7*b1 - 7) * q^53 + (-b3 - 3*b2) * q^55 + (-12*b1 + 12) * q^57 + (-2*b3 + 2*b2) * q^59 + 6*b1 * q^61 + 3*b3 * q^63 + (b1 - 3) * q^65 + 3*b3 * q^67 - 6 * q^69 + (-3*b3 + 3*b2) * q^71 + (7*b1 + 7) * q^73 + (4*b3 - 3*b2) * q^75 + (6*b1 + 6) * q^77 + 9 * q^81 + 7*b2 * q^83 + (-b1 + 3) * q^85 + 4*b2 * q^87 - 8*b1 * q^89 + (b3 + b2) * q^91 + (-6*b1 + 6) * q^93 + (-6*b3 + 2*b2) * q^95 + (7*b1 - 7) * q^97 + (3*b3 - 3*b2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 8 q^{5}+O(q^{10})$$ 4 * q + 8 * q^5 $$4 q + 8 q^{5} - 4 q^{13} + 4 q^{17} + 12 q^{25} - 16 q^{29} + 24 q^{33} - 20 q^{37} - 8 q^{41} - 12 q^{45} - 28 q^{53} + 48 q^{57} - 12 q^{65} - 24 q^{69} + 28 q^{73} + 24 q^{77} + 36 q^{81} + 12 q^{85} + 24 q^{93} - 28 q^{97}+O(q^{100})$$ 4 * q + 8 * q^5 - 4 * q^13 + 4 * q^17 + 12 * q^25 - 16 * q^29 + 24 * q^33 - 20 * q^37 - 8 * q^41 - 12 * q^45 - 28 * q^53 + 48 * q^57 - 12 * q^65 - 24 * q^69 + 28 * q^73 + 24 * q^77 + 36 * q^81 + 12 * q^85 + 24 * q^93 - 28 * q^97

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{12}^{3}$$ v^3 $$\beta_{2}$$ $$=$$ $$-\zeta_{12}^{3} + 2\zeta_{12}^{2} + 2\zeta_{12} - 1$$ -v^3 + 2*v^2 + 2*v - 1 $$\beta_{3}$$ $$=$$ $$-\zeta_{12}^{3} - 2\zeta_{12}^{2} + 2\zeta_{12} + 1$$ -v^3 - 2*v^2 + 2*v + 1
 $$\zeta_{12}$$ $$=$$ $$( \beta_{3} + \beta_{2} + 2\beta_1 ) / 4$$ (b3 + b2 + 2*b1) / 4 $$\zeta_{12}^{2}$$ $$=$$ $$( -\beta_{3} + \beta_{2} + 2 ) / 4$$ (-b3 + b2 + 2) / 4 $$\zeta_{12}^{3}$$ $$=$$ $$\beta_1$$ b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1280\mathbb{Z}\right)^\times$$.

 $$n$$ $$257$$ $$261$$ $$511$$ $$\chi(n)$$ $$-\beta_{1}$$ $$-1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
127.1
 0.866025 − 0.500000i −0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 + 0.500000i
0 −1.73205 + 1.73205i 0 2.00000 1.00000i 0 −1.73205 + 1.73205i 0 3.00000i 0
127.2 0 1.73205 1.73205i 0 2.00000 1.00000i 0 1.73205 1.73205i 0 3.00000i 0
383.1 0 −1.73205 1.73205i 0 2.00000 + 1.00000i 0 −1.73205 1.73205i 0 3.00000i 0
383.2 0 1.73205 + 1.73205i 0 2.00000 + 1.00000i 0 1.73205 + 1.73205i 0 3.00000i 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
4.b odd 2 1 inner
40.i odd 4 1 inner
40.k even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1280.2.o.r 4
4.b odd 2 1 inner 1280.2.o.r 4
5.c odd 4 1 1280.2.o.q 4
8.b even 2 1 1280.2.o.q 4
8.d odd 2 1 1280.2.o.q 4
16.e even 4 1 80.2.n.b 4
16.e even 4 1 320.2.n.i 4
16.f odd 4 1 80.2.n.b 4
16.f odd 4 1 320.2.n.i 4
20.e even 4 1 1280.2.o.q 4
40.i odd 4 1 inner 1280.2.o.r 4
40.k even 4 1 inner 1280.2.o.r 4
48.i odd 4 1 720.2.x.d 4
48.k even 4 1 720.2.x.d 4
80.i odd 4 1 80.2.n.b 4
80.i odd 4 1 1600.2.n.r 4
80.j even 4 1 320.2.n.i 4
80.j even 4 1 400.2.n.b 4
80.k odd 4 1 400.2.n.b 4
80.k odd 4 1 1600.2.n.r 4
80.q even 4 1 400.2.n.b 4
80.q even 4 1 1600.2.n.r 4
80.s even 4 1 80.2.n.b 4
80.s even 4 1 1600.2.n.r 4
80.t odd 4 1 320.2.n.i 4
80.t odd 4 1 400.2.n.b 4
240.t even 4 1 3600.2.x.e 4
240.z odd 4 1 720.2.x.d 4
240.bb even 4 1 720.2.x.d 4
240.bd odd 4 1 3600.2.x.e 4
240.bf even 4 1 3600.2.x.e 4
240.bm odd 4 1 3600.2.x.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.2.n.b 4 16.e even 4 1
80.2.n.b 4 16.f odd 4 1
80.2.n.b 4 80.i odd 4 1
80.2.n.b 4 80.s even 4 1
320.2.n.i 4 16.e even 4 1
320.2.n.i 4 16.f odd 4 1
320.2.n.i 4 80.j even 4 1
320.2.n.i 4 80.t odd 4 1
400.2.n.b 4 80.j even 4 1
400.2.n.b 4 80.k odd 4 1
400.2.n.b 4 80.q even 4 1
400.2.n.b 4 80.t odd 4 1
720.2.x.d 4 48.i odd 4 1
720.2.x.d 4 48.k even 4 1
720.2.x.d 4 240.z odd 4 1
720.2.x.d 4 240.bb even 4 1
1280.2.o.q 4 5.c odd 4 1
1280.2.o.q 4 8.b even 2 1
1280.2.o.q 4 8.d odd 2 1
1280.2.o.q 4 20.e even 4 1
1280.2.o.r 4 1.a even 1 1 trivial
1280.2.o.r 4 4.b odd 2 1 inner
1280.2.o.r 4 40.i odd 4 1 inner
1280.2.o.r 4 40.k even 4 1 inner
1600.2.n.r 4 80.i odd 4 1
1600.2.n.r 4 80.k odd 4 1
1600.2.n.r 4 80.q even 4 1
1600.2.n.r 4 80.s even 4 1
3600.2.x.e 4 240.t even 4 1
3600.2.x.e 4 240.bd odd 4 1
3600.2.x.e 4 240.bf even 4 1
3600.2.x.e 4 240.bm odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1280, [\chi])$$:

 $$T_{3}^{4} + 36$$ T3^4 + 36 $$T_{7}^{4} + 36$$ T7^4 + 36 $$T_{13}^{2} + 2T_{13} + 2$$ T13^2 + 2*T13 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + 36$$
$5$ $$(T^{2} - 4 T + 5)^{2}$$
$7$ $$T^{4} + 36$$
$11$ $$(T^{2} - 12)^{2}$$
$13$ $$(T^{2} + 2 T + 2)^{2}$$
$17$ $$(T^{2} - 2 T + 2)^{2}$$
$19$ $$(T^{2} + 48)^{2}$$
$23$ $$T^{4} + 36$$
$29$ $$(T + 4)^{4}$$
$31$ $$(T^{2} + 12)^{2}$$
$37$ $$(T^{2} + 10 T + 50)^{2}$$
$41$ $$(T + 2)^{4}$$
$43$ $$T^{4} + 36$$
$47$ $$T^{4} + 36$$
$53$ $$(T^{2} + 14 T + 98)^{2}$$
$59$ $$(T^{2} + 48)^{2}$$
$61$ $$(T^{2} + 36)^{2}$$
$67$ $$T^{4} + 2916$$
$71$ $$(T^{2} + 108)^{2}$$
$73$ $$(T^{2} - 14 T + 98)^{2}$$
$79$ $$T^{4}$$
$83$ $$T^{4} + 86436$$
$89$ $$(T^{2} + 64)^{2}$$
$97$ $$(T^{2} + 14 T + 98)^{2}$$