# Properties

 Label 1280.4.d.w 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: $$\Q(i, \sqrt{21})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 11x^{2} + 25$$ x^4 + 11*x^2 + 25 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}$$ 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.

 $$f(q)$$ $$=$$ $$q + (\beta_{2} - \beta_1) q^{3} - 5 \beta_1 q^{5} + (3 \beta_{3} + 3) q^{7} + (2 \beta_{3} + 5) q^{9}+O(q^{10})$$ q + (b2 - b1) * q^3 - 5*b1 * q^5 + (3*b3 + 3) * q^7 + (2*b3 + 5) * q^9 $$q + (\beta_{2} - \beta_1) q^{3} - 5 \beta_1 q^{5} + (3 \beta_{3} + 3) q^{7} + (2 \beta_{3} + 5) q^{9} - 14 \beta_1 q^{11} + (10 \beta_{2} - 16 \beta_1) q^{13} + (5 \beta_{3} - 5) q^{15} + ( - 14 \beta_{3} - 48) q^{17} - 2 \beta_{2} q^{19} + 60 \beta_1 q^{21} + ( - 3 \beta_{3} - 75) q^{23} - 25 q^{25} + (30 \beta_{2} + 10 \beta_1) q^{27} + (32 \beta_{2} - 86 \beta_1) q^{29} + ( - 42 \beta_{3} - 74) q^{31} + (14 \beta_{3} - 14) q^{33} + ( - 15 \beta_{2} - 15 \beta_1) q^{35} + (20 \beta_{2} + 22 \beta_1) q^{37} + (26 \beta_{3} - 226) q^{39} + (62 \beta_{3} + 120) q^{41} + ( - 5 \beta_{2} + 137 \beta_1) q^{43} + ( - 10 \beta_{2} - 25 \beta_1) q^{45} + ( - 11 \beta_{3} + 201) q^{47} + (18 \beta_{3} - 145) q^{49} + ( - 34 \beta_{2} - 246 \beta_1) q^{51} + (58 \beta_{2} + 188 \beta_1) q^{53} - 70 q^{55} + ( - 2 \beta_{3} + 42) q^{57} + ( - 46 \beta_{2} + 516 \beta_1) q^{59} + (96 \beta_{2} - 306 \beta_1) q^{61} + (21 \beta_{3} + 141) q^{63} + (50 \beta_{3} - 80) q^{65} + ( - 45 \beta_{2} - 47 \beta_1) q^{67} + ( - 72 \beta_{2} + 12 \beta_1) q^{69} + ( - 58 \beta_{3} - 518) q^{71} + ( - 2 \beta_{3} + 400) q^{73} + ( - 25 \beta_{2} + 25 \beta_1) q^{75} + ( - 42 \beta_{2} - 42 \beta_1) q^{77} + (172 \beta_{3} - 52) q^{79} + (74 \beta_{3} - 485) q^{81} + (253 \beta_{2} - 29 \beta_1) q^{83} + (70 \beta_{2} + 240 \beta_1) q^{85} + (118 \beta_{3} - 758) q^{87} + (132 \beta_{3} + 214) q^{89} + ( - 18 \beta_{2} + 582 \beta_1) q^{91} + ( - 32 \beta_{2} - 808 \beta_1) q^{93} - 10 \beta_{3} q^{95} + (206 \beta_{3} - 732) q^{97} + ( - 28 \beta_{2} - 70 \beta_1) q^{99}+O(q^{100})$$ q + (b2 - b1) * q^3 - 5*b1 * q^5 + (3*b3 + 3) * q^7 + (2*b3 + 5) * q^9 - 14*b1 * q^11 + (10*b2 - 16*b1) * q^13 + (5*b3 - 5) * q^15 + (-14*b3 - 48) * q^17 - 2*b2 * q^19 + 60*b1 * q^21 + (-3*b3 - 75) * q^23 - 25 * q^25 + (30*b2 + 10*b1) * q^27 + (32*b2 - 86*b1) * q^29 + (-42*b3 - 74) * q^31 + (14*b3 - 14) * q^33 + (-15*b2 - 15*b1) * q^35 + (20*b2 + 22*b1) * q^37 + (26*b3 - 226) * q^39 + (62*b3 + 120) * q^41 + (-5*b2 + 137*b1) * q^43 + (-10*b2 - 25*b1) * q^45 + (-11*b3 + 201) * q^47 + (18*b3 - 145) * q^49 + (-34*b2 - 246*b1) * q^51 + (58*b2 + 188*b1) * q^53 - 70 * q^55 + (-2*b3 + 42) * q^57 + (-46*b2 + 516*b1) * q^59 + (96*b2 - 306*b1) * q^61 + (21*b3 + 141) * q^63 + (50*b3 - 80) * q^65 + (-45*b2 - 47*b1) * q^67 + (-72*b2 + 12*b1) * q^69 + (-58*b3 - 518) * q^71 + (-2*b3 + 400) * q^73 + (-25*b2 + 25*b1) * q^75 + (-42*b2 - 42*b1) * q^77 + (172*b3 - 52) * q^79 + (74*b3 - 485) * q^81 + (253*b2 - 29*b1) * q^83 + (70*b2 + 240*b1) * q^85 + (118*b3 - 758) * q^87 + (132*b3 + 214) * q^89 + (-18*b2 + 582*b1) * q^91 + (-32*b2 - 808*b1) * q^93 - 10*b3 * q^95 + (206*b3 - 732) * q^97 + (-28*b2 - 70*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 12 q^{7} + 20 q^{9}+O(q^{10})$$ 4 * q + 12 * q^7 + 20 * q^9 $$4 q + 12 q^{7} + 20 q^{9} - 20 q^{15} - 192 q^{17} - 300 q^{23} - 100 q^{25} - 296 q^{31} - 56 q^{33} - 904 q^{39} + 480 q^{41} + 804 q^{47} - 580 q^{49} - 280 q^{55} + 168 q^{57} + 564 q^{63} - 320 q^{65} - 2072 q^{71} + 1600 q^{73} - 208 q^{79} - 1940 q^{81} - 3032 q^{87} + 856 q^{89} - 2928 q^{97}+O(q^{100})$$ 4 * q + 12 * q^7 + 20 * q^9 - 20 * q^15 - 192 * q^17 - 300 * q^23 - 100 * q^25 - 296 * q^31 - 56 * q^33 - 904 * q^39 + 480 * q^41 + 804 * q^47 - 580 * q^49 - 280 * q^55 + 168 * q^57 + 564 * q^63 - 320 * q^65 - 2072 * q^71 + 1600 * q^73 - 208 * q^79 - 1940 * q^81 - 3032 * q^87 + 856 * q^89 - 2928 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 11x^{2} + 25$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} + 6\nu ) / 5$$ (v^3 + 6*v) / 5 $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 16\nu ) / 5$$ (v^3 + 16*v) / 5 $$\beta_{3}$$ $$=$$ $$2\nu^{2} + 11$$ 2*v^2 + 11
 $$\nu$$ $$=$$ $$( \beta_{2} - \beta_1 ) / 2$$ (b2 - b1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{3} - 11 ) / 2$$ (b3 - 11) / 2 $$\nu^{3}$$ $$=$$ $$-3\beta_{2} + 8\beta_1$$ -3*b2 + 8*b1

## Character values

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

 $$n$$ $$257$$ $$261$$ $$511$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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
 − 2.79129i − 1.79129i 1.79129i 2.79129i
0 5.58258i 0 5.00000i 0 −10.7477 0 −4.16515 0
641.2 0 3.58258i 0 5.00000i 0 16.7477 0 14.1652 0
641.3 0 3.58258i 0 5.00000i 0 16.7477 0 14.1652 0
641.4 0 5.58258i 0 5.00000i 0 −10.7477 0 −4.16515 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.w 4
4.b odd 2 1 1280.4.d.s 4
8.b even 2 1 inner 1280.4.d.w 4
8.d odd 2 1 1280.4.d.s 4
16.e even 4 1 640.4.a.h yes 2
16.e even 4 1 640.4.a.i yes 2
16.f odd 4 1 640.4.a.g 2
16.f odd 4 1 640.4.a.j yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
640.4.a.g 2 16.f odd 4 1
640.4.a.h yes 2 16.e even 4 1
640.4.a.i yes 2 16.e even 4 1
640.4.a.j yes 2 16.f odd 4 1
1280.4.d.s 4 4.b odd 2 1
1280.4.d.s 4 8.d odd 2 1
1280.4.d.w 4 1.a even 1 1 trivial
1280.4.d.w 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 $$S_{4}^{\mathrm{new}}(1280, [\chi])$$:

 $$T_{3}^{4} + 44T_{3}^{2} + 400$$ T3^4 + 44*T3^2 + 400 $$T_{7}^{2} - 6T_{7} - 180$$ T7^2 - 6*T7 - 180 $$T_{11}^{2} + 196$$ T11^2 + 196

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + 44T^{2} + 400$$
$5$ $$(T^{2} + 25)^{2}$$
$7$ $$(T^{2} - 6 T - 180)^{2}$$
$11$ $$(T^{2} + 196)^{2}$$
$13$ $$T^{4} + 4712 T^{2} + \cdots + 3400336$$
$17$ $$(T^{2} + 96 T - 1812)^{2}$$
$19$ $$(T^{2} + 84)^{2}$$
$23$ $$(T^{2} + 150 T + 5436)^{2}$$
$29$ $$T^{4} + 57800 T^{2} + \cdots + 199035664$$
$31$ $$(T^{2} + 148 T - 31568)^{2}$$
$37$ $$T^{4} + 17768 T^{2} + \cdots + 62663056$$
$41$ $$(T^{2} - 240 T - 66324)^{2}$$
$43$ $$T^{4} + 38588 T^{2} + \cdots + 332843536$$
$47$ $$(T^{2} - 402 T + 37860)^{2}$$
$53$ $$T^{4} + 211976 T^{2} + \cdots + 1246090000$$
$59$ $$T^{4} + 621384 T^{2} + \cdots + 49204112400$$
$61$ $$T^{4} + 574344 T^{2} + \cdots + 9980010000$$
$67$ $$T^{4} + 89468 T^{2} + \cdots + 1625379856$$
$71$ $$(T^{2} + 1036 T + 197680)^{2}$$
$73$ $$(T^{2} - 800 T + 159916)^{2}$$
$79$ $$(T^{2} + 104 T - 618560)^{2}$$
$83$ $$T^{4} + 2690060 T^{2} + \cdots + 1804583849104$$
$89$ $$(T^{2} - 428 T - 320108)^{2}$$
$97$ $$(T^{2} + 1464 T - 355332)^{2}$$