# Properties

 Label 1280.4.d.y Level $1280$ Weight $4$ Character orbit 1280.d Analytic conductor $75.522$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Coefficient field: $$\Q(i, \sqrt{51})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 25x^{2} + 169$$ x^4 - 25*x^2 + 169 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{6}$$ Twist minimal: no (minimal twist has level 640) 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 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.

 $$f(q)$$ $$=$$ $$q + 2 \beta_1 q^{3} + 5 \beta_1 q^{5} + ( - \beta_{2} + 4) q^{7} + 23 q^{9}+O(q^{10})$$ q + 2*b1 * q^3 + 5*b1 * q^5 + (-b2 + 4) * q^7 + 23 * q^9 $$q + 2 \beta_1 q^{3} + 5 \beta_1 q^{5} + ( - \beta_{2} + 4) q^{7} + 23 q^{9} + ( - \beta_{3} + 18 \beta_1) q^{11} + ( - 2 \beta_{3} + 2 \beta_1) q^{13} - 10 q^{15} + (2 \beta_{2} - 2) q^{17} + (\beta_{3} + 34 \beta_1) q^{19} + ( - 2 \beta_{3} + 8 \beta_1) q^{21} + (\beta_{2} - 132) q^{23} - 25 q^{25} + 100 \beta_1 q^{27} + ( - 2 \beta_{3} + 170 \beta_1) q^{29} + ( - 2 \beta_{2} + 160) q^{31} + (2 \beta_{2} - 36) q^{33} + ( - 5 \beta_{3} + 20 \beta_1) q^{35} + ( - 8 \beta_{3} - 106 \beta_1) q^{37} + (4 \beta_{2} - 4) q^{39} + (12 \beta_{2} - 90) q^{41} + (10 \beta_{3} + 138 \beta_1) q^{43} + 115 \beta_1 q^{45} + (11 \beta_{2} + 24) q^{47} + ( - 8 \beta_{2} + 489) q^{49} + (4 \beta_{3} - 4 \beta_1) q^{51} + (14 \beta_{3} - 106 \beta_1) q^{53} + (5 \beta_{2} - 90) q^{55} + ( - 2 \beta_{2} - 68) q^{57} + ( - 11 \beta_{3} + 162 \beta_1) q^{59} + (24 \beta_{3} - 78 \beta_1) q^{61} + ( - 23 \beta_{2} + 92) q^{63} + (10 \beta_{2} - 10) q^{65} + (10 \beta_{3} - 118 \beta_1) q^{67} + (2 \beta_{3} - 264 \beta_1) q^{69} + ( - 24 \beta_{2} - 364) q^{71} + ( - 26 \beta_{2} - 94) q^{73} - 50 \beta_1 q^{75} + ( - 22 \beta_{3} + 888 \beta_1) q^{77} + (20 \beta_{2} + 24) q^{79} + 421 q^{81} + ( - 4 \beta_{3} - 582 \beta_1) q^{83} + (10 \beta_{3} - 10 \beta_1) q^{85} + (4 \beta_{2} - 340) q^{87} + ( - 24 \beta_{2} - 702) q^{89} + ( - 10 \beta_{3} + 1640 \beta_1) q^{91} + ( - 4 \beta_{3} + 320 \beta_1) q^{93} + ( - 5 \beta_{2} - 170) q^{95} + ( - 38 \beta_{2} + 574) q^{97} + ( - 23 \beta_{3} + 414 \beta_1) q^{99}+O(q^{100})$$ q + 2*b1 * q^3 + 5*b1 * q^5 + (-b2 + 4) * q^7 + 23 * q^9 + (-b3 + 18*b1) * q^11 + (-2*b3 + 2*b1) * q^13 - 10 * q^15 + (2*b2 - 2) * q^17 + (b3 + 34*b1) * q^19 + (-2*b3 + 8*b1) * q^21 + (b2 - 132) * q^23 - 25 * q^25 + 100*b1 * q^27 + (-2*b3 + 170*b1) * q^29 + (-2*b2 + 160) * q^31 + (2*b2 - 36) * q^33 + (-5*b3 + 20*b1) * q^35 + (-8*b3 - 106*b1) * q^37 + (4*b2 - 4) * q^39 + (12*b2 - 90) * q^41 + (10*b3 + 138*b1) * q^43 + 115*b1 * q^45 + (11*b2 + 24) * q^47 + (-8*b2 + 489) * q^49 + (4*b3 - 4*b1) * q^51 + (14*b3 - 106*b1) * q^53 + (5*b2 - 90) * q^55 + (-2*b2 - 68) * q^57 + (-11*b3 + 162*b1) * q^59 + (24*b3 - 78*b1) * q^61 + (-23*b2 + 92) * q^63 + (10*b2 - 10) * q^65 + (10*b3 - 118*b1) * q^67 + (2*b3 - 264*b1) * q^69 + (-24*b2 - 364) * q^71 + (-26*b2 - 94) * q^73 - 50*b1 * q^75 + (-22*b3 + 888*b1) * q^77 + (20*b2 + 24) * q^79 + 421 * q^81 + (-4*b3 - 582*b1) * q^83 + (10*b3 - 10*b1) * q^85 + (4*b2 - 340) * q^87 + (-24*b2 - 702) * q^89 + (-10*b3 + 1640*b1) * q^91 + (-4*b3 + 320*b1) * q^93 + (-5*b2 - 170) * q^95 + (-38*b2 + 574) * q^97 + (-23*b3 + 414*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 16 q^{7} + 92 q^{9}+O(q^{10})$$ 4 * q + 16 * q^7 + 92 * q^9 $$4 q + 16 q^{7} + 92 q^{9} - 40 q^{15} - 8 q^{17} - 528 q^{23} - 100 q^{25} + 640 q^{31} - 144 q^{33} - 16 q^{39} - 360 q^{41} + 96 q^{47} + 1956 q^{49} - 360 q^{55} - 272 q^{57} + 368 q^{63} - 40 q^{65} - 1456 q^{71} - 376 q^{73} + 96 q^{79} + 1684 q^{81} - 1360 q^{87} - 2808 q^{89} - 680 q^{95} + 2296 q^{97}+O(q^{100})$$ 4 * q + 16 * q^7 + 92 * q^9 - 40 * q^15 - 8 * q^17 - 528 * q^23 - 100 * q^25 + 640 * q^31 - 144 * q^33 - 16 * q^39 - 360 * q^41 + 96 * q^47 + 1956 * q^49 - 360 * q^55 - 272 * q^57 + 368 * q^63 - 40 * q^65 - 1456 * q^71 - 376 * q^73 + 96 * q^79 + 1684 * q^81 - 1360 * q^87 - 2808 * q^89 - 680 * q^95 + 2296 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 25x^{2} + 169$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} - 12\nu ) / 13$$ (v^3 - 12*v) / 13 $$\beta_{2}$$ $$=$$ $$( -4\nu^{3} + 152\nu ) / 13$$ (-4*v^3 + 152*v) / 13 $$\beta_{3}$$ $$=$$ $$8\nu^{2} - 100$$ 8*v^2 - 100
 $$\nu$$ $$=$$ $$( \beta_{2} + 4\beta_1 ) / 8$$ (b2 + 4*b1) / 8 $$\nu^{2}$$ $$=$$ $$( \beta_{3} + 100 ) / 8$$ (b3 + 100) / 8 $$\nu^{3}$$ $$=$$ $$( 3\beta_{2} + 38\beta_1 ) / 2$$ (3*b2 + 38*b1) / 2

## 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
 3.57071 − 0.500000i −3.57071 − 0.500000i 3.57071 + 0.500000i −3.57071 + 0.500000i
0 2.00000i 0 5.00000i 0 −24.5657 0 23.0000 0
641.2 0 2.00000i 0 5.00000i 0 32.5657 0 23.0000 0
641.3 0 2.00000i 0 5.00000i 0 −24.5657 0 23.0000 0
641.4 0 2.00000i 0 5.00000i 0 32.5657 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.y 4
4.b odd 2 1 1280.4.d.r 4
8.b even 2 1 inner 1280.4.d.y 4
8.d odd 2 1 1280.4.d.r 4
16.e even 4 1 640.4.a.f yes 2
16.e even 4 1 640.4.a.k yes 2
16.f odd 4 1 640.4.a.e 2
16.f odd 4 1 640.4.a.l yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
640.4.a.e 2 16.f odd 4 1
640.4.a.f yes 2 16.e even 4 1
640.4.a.k yes 2 16.e even 4 1
640.4.a.l yes 2 16.f odd 4 1
1280.4.d.r 4 4.b odd 2 1
1280.4.d.r 4 8.d odd 2 1
1280.4.d.y 4 1.a even 1 1 trivial
1280.4.d.y 4 8.b even 2 1 inner

## 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 $$T_{7}^{2} - 8T_{7} - 800$$ T7^2 - 8*T7 - 800 $$T_{11}^{4} + 2280T_{11}^{2} + 242064$$ T11^4 + 2280*T11^2 + 242064

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T^{2} + 4)^{2}$$
$5$ $$(T^{2} + 25)^{2}$$
$7$ $$(T^{2} - 8 T - 800)^{2}$$
$11$ $$T^{4} + 2280 T^{2} + 242064$$
$13$ $$T^{4} + 6536 T^{2} + \cdots + 10627600$$
$17$ $$(T^{2} + 4 T - 3260)^{2}$$
$19$ $$T^{4} + 3944 T^{2} + 115600$$
$23$ $$(T^{2} + 264 T + 16608)^{2}$$
$29$ $$T^{4} + 64328 T^{2} + \cdots + 657204496$$
$31$ $$(T^{2} - 320 T + 22336)^{2}$$
$37$ $$T^{4} + 126920 T^{2} + \cdots + 1680016144$$
$41$ $$(T^{2} + 180 T - 109404)^{2}$$
$43$ $$T^{4} + 201288 T^{2} + \cdots + 3913253136$$
$47$ $$(T^{2} - 48 T - 98160)^{2}$$
$53$ $$T^{4} + 342344 T^{2} + \cdots + 22111690000$$
$59$ $$T^{4} + 249960 T^{2} + \cdots + 5255090064$$
$61$ $$T^{4} + 952200 T^{2} + \cdots + 215232900624$$
$67$ $$T^{4} + 191048 T^{2} + \cdots + 4580040976$$
$71$ $$(T^{2} + 728 T - 337520)^{2}$$
$73$ $$(T^{2} + 188 T - 542780)^{2}$$
$79$ $$(T^{2} - 48 T - 325824)^{2}$$
$83$ $$T^{4} + 703560 T^{2} + \cdots + 106059646224$$
$89$ $$(T^{2} + 1404 T + 22788)^{2}$$
$97$ $$(T^{2} - 1148 T - 848828)^{2}$$