# Properties

 Label 1280.4.d.q 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: $$\Q(i, \sqrt{6})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 9$$ x^4 + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 160) 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{2} + 4 \beta_1) q^{3} - 5 \beta_1 q^{5} + ( - 5 \beta_{3} - 4) q^{7} + (8 \beta_{3} - 13) q^{9}+O(q^{10})$$ q + (-b2 + 4*b1) * q^3 - 5*b1 * q^5 + (-5*b3 - 4) * q^7 + (8*b3 - 13) * q^9 $$q + ( - \beta_{2} + 4 \beta_1) q^{3} - 5 \beta_1 q^{5} + ( - 5 \beta_{3} - 4) q^{7} + (8 \beta_{3} - 13) q^{9} + (6 \beta_{2} - 32 \beta_1) q^{11} + ( - 8 \beta_{2} + 6 \beta_1) q^{13} + ( - 5 \beta_{3} + 20) q^{15} + (24 \beta_{3} + 2) q^{17} + (8 \beta_{2} + 104 \beta_1) q^{19} + ( - 16 \beta_{2} + 104 \beta_1) q^{21} + ( - 11 \beta_{3} - 60) q^{23} - 25 q^{25} + (18 \beta_{2} - 136 \beta_1) q^{27} + (16 \beta_{2} - 146 \beta_1) q^{29} + (6 \beta_{3} + 88) q^{31} + ( - 56 \beta_{3} + 272) q^{33} + (25 \beta_{2} + 20 \beta_1) q^{35} + (16 \beta_{2} + 178 \beta_1) q^{37} + (38 \beta_{3} - 216) q^{39} + ( - 40 \beta_{3} - 50) q^{41} + (37 \beta_{2} + 188 \beta_1) q^{43} + ( - 40 \beta_{2} + 65 \beta_1) q^{45} + (13 \beta_{3} - 140) q^{47} + (40 \beta_{3} + 273) q^{49} + (94 \beta_{2} - 568 \beta_1) q^{51} + (24 \beta_{2} - 158 \beta_1) q^{53} + (30 \beta_{3} - 160) q^{55} + (72 \beta_{3} - 224) q^{57} + ( - 20 \beta_{2} - 360 \beta_1) q^{59} + (32 \beta_{2} - 634 \beta_1) q^{61} + (33 \beta_{3} - 908) q^{63} + ( - 40 \beta_{3} + 30) q^{65} + ( - 47 \beta_{2} - 372 \beta_1) q^{67} + (16 \beta_{2} + 24 \beta_1) q^{69} + (218 \beta_{3} + 24) q^{71} + ( - 120 \beta_{3} + 470) q^{73} + (25 \beta_{2} - 100 \beta_1) q^{75} + (136 \beta_{2} - 592 \beta_1) q^{77} + (172 \beta_{3} + 16) q^{79} + (8 \beta_{3} + 625) q^{81} + (47 \beta_{2} - 796 \beta_1) q^{83} + ( - 120 \beta_{2} - 10 \beta_1) q^{85} + ( - 210 \beta_{3} + 968) q^{87} + (48 \beta_{3} + 390) q^{89} + (2 \beta_{2} + 936 \beta_1) q^{91} + ( - 64 \beta_{2} + 208 \beta_1) q^{93} + (40 \beta_{3} + 520) q^{95} + (40 \beta_{3} + 610) q^{97} + ( - 334 \beta_{2} + 1568 \beta_1) q^{99}+O(q^{100})$$ q + (-b2 + 4*b1) * q^3 - 5*b1 * q^5 + (-5*b3 - 4) * q^7 + (8*b3 - 13) * q^9 + (6*b2 - 32*b1) * q^11 + (-8*b2 + 6*b1) * q^13 + (-5*b3 + 20) * q^15 + (24*b3 + 2) * q^17 + (8*b2 + 104*b1) * q^19 + (-16*b2 + 104*b1) * q^21 + (-11*b3 - 60) * q^23 - 25 * q^25 + (18*b2 - 136*b1) * q^27 + (16*b2 - 146*b1) * q^29 + (6*b3 + 88) * q^31 + (-56*b3 + 272) * q^33 + (25*b2 + 20*b1) * q^35 + (16*b2 + 178*b1) * q^37 + (38*b3 - 216) * q^39 + (-40*b3 - 50) * q^41 + (37*b2 + 188*b1) * q^43 + (-40*b2 + 65*b1) * q^45 + (13*b3 - 140) * q^47 + (40*b3 + 273) * q^49 + (94*b2 - 568*b1) * q^51 + (24*b2 - 158*b1) * q^53 + (30*b3 - 160) * q^55 + (72*b3 - 224) * q^57 + (-20*b2 - 360*b1) * q^59 + (32*b2 - 634*b1) * q^61 + (33*b3 - 908) * q^63 + (-40*b3 + 30) * q^65 + (-47*b2 - 372*b1) * q^67 + (16*b2 + 24*b1) * q^69 + (218*b3 + 24) * q^71 + (-120*b3 + 470) * q^73 + (25*b2 - 100*b1) * q^75 + (136*b2 - 592*b1) * q^77 + (172*b3 + 16) * q^79 + (8*b3 + 625) * q^81 + (47*b2 - 796*b1) * q^83 + (-120*b2 - 10*b1) * q^85 + (-210*b3 + 968) * q^87 + (48*b3 + 390) * q^89 + (2*b2 + 936*b1) * q^91 + (-64*b2 + 208*b1) * q^93 + (40*b3 + 520) * q^95 + (40*b3 + 610) * q^97 + (-334*b2 + 1568*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 16 q^{7} - 52 q^{9}+O(q^{10})$$ 4 * q - 16 * q^7 - 52 * q^9 $$4 q - 16 q^{7} - 52 q^{9} + 80 q^{15} + 8 q^{17} - 240 q^{23} - 100 q^{25} + 352 q^{31} + 1088 q^{33} - 864 q^{39} - 200 q^{41} - 560 q^{47} + 1092 q^{49} - 640 q^{55} - 896 q^{57} - 3632 q^{63} + 120 q^{65} + 96 q^{71} + 1880 q^{73} + 64 q^{79} + 2500 q^{81} + 3872 q^{87} + 1560 q^{89} + 2080 q^{95} + 2440 q^{97}+O(q^{100})$$ 4 * q - 16 * q^7 - 52 * q^9 + 80 * q^15 + 8 * q^17 - 240 * q^23 - 100 * q^25 + 352 * q^31 + 1088 * q^33 - 864 * q^39 - 200 * q^41 - 560 * q^47 + 1092 * q^49 - 640 * q^55 - 896 * q^57 - 3632 * q^63 + 120 * q^65 + 96 * q^71 + 1880 * q^73 + 64 * q^79 + 2500 * q^81 + 3872 * q^87 + 1560 * q^89 + 2080 * q^95 + 2440 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 9$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{2} ) / 3$$ (v^2) / 3 $$\beta_{2}$$ $$=$$ $$( 2\nu^{3} + 6\nu ) / 3$$ (2*v^3 + 6*v) / 3 $$\beta_{3}$$ $$=$$ $$( -2\nu^{3} + 6\nu ) / 3$$ (-2*v^3 + 6*v) / 3
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_{2} ) / 4$$ (b3 + b2) / 4 $$\nu^{2}$$ $$=$$ $$3\beta_1$$ 3*b1 $$\nu^{3}$$ $$=$$ $$( -3\beta_{3} + 3\beta_{2} ) / 4$$ (-3*b3 + 3*b2) / 4

## 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
 −1.22474 + 1.22474i 1.22474 + 1.22474i 1.22474 − 1.22474i −1.22474 − 1.22474i
0 8.89898i 0 5.00000i 0 20.4949 0 −52.1918 0
641.2 0 0.898979i 0 5.00000i 0 −28.4949 0 26.1918 0
641.3 0 0.898979i 0 5.00000i 0 −28.4949 0 26.1918 0
641.4 0 8.89898i 0 5.00000i 0 20.4949 0 −52.1918 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.q 4
4.b odd 2 1 1280.4.d.x 4
8.b even 2 1 inner 1280.4.d.q 4
8.d odd 2 1 1280.4.d.x 4
16.e even 4 1 160.4.a.g yes 2
16.e even 4 1 320.4.a.o 2
16.f odd 4 1 160.4.a.c 2
16.f odd 4 1 320.4.a.s 2
48.i odd 4 1 1440.4.a.x 2
48.k even 4 1 1440.4.a.t 2
80.i odd 4 1 800.4.c.k 4
80.j even 4 1 800.4.c.i 4
80.k odd 4 1 800.4.a.s 2
80.k odd 4 1 1600.4.a.cd 2
80.q even 4 1 800.4.a.m 2
80.q even 4 1 1600.4.a.cn 2
80.s even 4 1 800.4.c.i 4
80.t odd 4 1 800.4.c.k 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.4.a.c 2 16.f odd 4 1
160.4.a.g yes 2 16.e even 4 1
320.4.a.o 2 16.e even 4 1
320.4.a.s 2 16.f odd 4 1
800.4.a.m 2 80.q even 4 1
800.4.a.s 2 80.k odd 4 1
800.4.c.i 4 80.j even 4 1
800.4.c.i 4 80.s even 4 1
800.4.c.k 4 80.i odd 4 1
800.4.c.k 4 80.t odd 4 1
1280.4.d.q 4 1.a even 1 1 trivial
1280.4.d.q 4 8.b even 2 1 inner
1280.4.d.x 4 4.b odd 2 1
1280.4.d.x 4 8.d odd 2 1
1440.4.a.t 2 48.k even 4 1
1440.4.a.x 2 48.i odd 4 1
1600.4.a.cd 2 80.k odd 4 1
1600.4.a.cn 2 80.q even 4 1

## Hecke kernels

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

 $$T_{3}^{4} + 80T_{3}^{2} + 64$$ T3^4 + 80*T3^2 + 64 $$T_{7}^{2} + 8T_{7} - 584$$ T7^2 + 8*T7 - 584 $$T_{11}^{4} + 3776T_{11}^{2} + 25600$$ T11^4 + 3776*T11^2 + 25600

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + 80T^{2} + 64$$
$5$ $$(T^{2} + 25)^{2}$$
$7$ $$(T^{2} + 8 T - 584)^{2}$$
$11$ $$T^{4} + 3776 T^{2} + 25600$$
$13$ $$T^{4} + 3144 T^{2} + \cdots + 2250000$$
$17$ $$(T^{2} - 4 T - 13820)^{2}$$
$19$ $$T^{4} + 24704 T^{2} + \cdots + 86118400$$
$23$ $$(T^{2} + 120 T + 696)^{2}$$
$29$ $$T^{4} + 54920 T^{2} + \cdots + 230189584$$
$31$ $$(T^{2} - 176 T + 6880)^{2}$$
$37$ $$T^{4} + 75656 T^{2} + \cdots + 652291600$$
$41$ $$(T^{2} + 100 T - 35900)^{2}$$
$43$ $$T^{4} + 136400 T^{2} + \cdots + 6190144$$
$47$ $$(T^{2} + 280 T + 15544)^{2}$$
$53$ $$T^{4} + 77576 T^{2} + \cdots + 124099600$$
$59$ $$T^{4} + 278400 T^{2} + \cdots + 14400000000$$
$61$ $$T^{4} + 853064 T^{2} + \cdots + 142415664400$$
$67$ $$T^{4} + 382800 T^{2} + \cdots + 7287695424$$
$71$ $$(T^{2} - 48 T - 1140000)^{2}$$
$73$ $$(T^{2} - 940 T - 124700)^{2}$$
$79$ $$(T^{2} - 32 T - 709760)^{2}$$
$83$ $$T^{4} + 1373264 T^{2} + \cdots + 337096360000$$
$89$ $$(T^{2} - 780 T + 96804)^{2}$$
$97$ $$(T^{2} - 1220 T + 333700)^{2}$$