# Properties

 Label 800.4.c.a Level $800$ Weight $4$ Character orbit 800.c Analytic conductor $47.202$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [800,4,Mod(449,800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(800, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("800.449");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$800 = 2^{5} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 800.c (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: $$47.2015280046$$ 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: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 32) 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 $$\beta = 2i$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 4 \beta q^{3} - 8 \beta q^{7} - 37 q^{9} +O(q^{10})$$ q + 4*b * q^3 - 8*b * q^7 - 37 * q^9 $$q + 4 \beta q^{3} - 8 \beta q^{7} - 37 q^{9} - 40 q^{11} - 25 \beta q^{13} + 15 \beta q^{17} - 40 q^{19} + 128 q^{21} + 24 \beta q^{23} - 40 \beta q^{27} + 34 q^{29} + 320 q^{31} - 160 \beta q^{33} - 155 \beta q^{37} + 400 q^{39} + 410 q^{41} + 76 \beta q^{43} + 208 \beta q^{47} + 87 q^{49} - 240 q^{51} - 205 \beta q^{53} - 160 \beta q^{57} + 200 q^{59} + 30 q^{61} + 296 \beta q^{63} - 388 \beta q^{67} - 384 q^{69} + 400 q^{71} - 315 \beta q^{73} + 320 \beta q^{77} + 1120 q^{79} - 359 q^{81} + 276 \beta q^{83} + 136 \beta q^{87} + 326 q^{89} - 800 q^{91} + 1280 \beta q^{93} + 55 \beta q^{97} + 1480 q^{99} +O(q^{100})$$ q + 4*b * q^3 - 8*b * q^7 - 37 * q^9 - 40 * q^11 - 25*b * q^13 + 15*b * q^17 - 40 * q^19 + 128 * q^21 + 24*b * q^23 - 40*b * q^27 + 34 * q^29 + 320 * q^31 - 160*b * q^33 - 155*b * q^37 + 400 * q^39 + 410 * q^41 + 76*b * q^43 + 208*b * q^47 + 87 * q^49 - 240 * q^51 - 205*b * q^53 - 160*b * q^57 + 200 * q^59 + 30 * q^61 + 296*b * q^63 - 388*b * q^67 - 384 * q^69 + 400 * q^71 - 315*b * q^73 + 320*b * q^77 + 1120 * q^79 - 359 * q^81 + 276*b * q^83 + 136*b * q^87 + 326 * q^89 - 800 * q^91 + 1280*b * q^93 + 55*b * q^97 + 1480 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 74 q^{9}+O(q^{10})$$ 2 * q - 74 * q^9 $$2 q - 74 q^{9} - 80 q^{11} - 80 q^{19} + 256 q^{21} + 68 q^{29} + 640 q^{31} + 800 q^{39} + 820 q^{41} + 174 q^{49} - 480 q^{51} + 400 q^{59} + 60 q^{61} - 768 q^{69} + 800 q^{71} + 2240 q^{79} - 718 q^{81} + 652 q^{89} - 1600 q^{91} + 2960 q^{99}+O(q^{100})$$ 2 * q - 74 * q^9 - 80 * q^11 - 80 * q^19 + 256 * q^21 + 68 * q^29 + 640 * q^31 + 800 * q^39 + 820 * q^41 + 174 * q^49 - 480 * q^51 + 400 * q^59 + 60 * q^61 - 768 * q^69 + 800 * q^71 + 2240 * q^79 - 718 * q^81 + 652 * q^89 - 1600 * q^91 + 2960 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/800\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$351$$ $$577$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
449.1
 − 1.00000i 1.00000i
0 8.00000i 0 0 0 16.0000i 0 −37.0000 0
449.2 0 8.00000i 0 0 0 16.0000i 0 −37.0000 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
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 800.4.c.a 2
4.b odd 2 1 800.4.c.b 2
5.b even 2 1 inner 800.4.c.a 2
5.c odd 4 1 32.4.a.c yes 1
5.c odd 4 1 800.4.a.a 1
15.e even 4 1 288.4.a.i 1
20.d odd 2 1 800.4.c.b 2
20.e even 4 1 32.4.a.a 1
20.e even 4 1 800.4.a.k 1
35.f even 4 1 1568.4.a.c 1
40.i odd 4 1 64.4.a.a 1
40.i odd 4 1 1600.4.a.bw 1
40.k even 4 1 64.4.a.e 1
40.k even 4 1 1600.4.a.e 1
60.l odd 4 1 288.4.a.h 1
80.i odd 4 1 256.4.b.c 2
80.j even 4 1 256.4.b.e 2
80.s even 4 1 256.4.b.e 2
80.t odd 4 1 256.4.b.c 2
120.q odd 4 1 576.4.a.g 1
120.w even 4 1 576.4.a.h 1
140.j odd 4 1 1568.4.a.o 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
32.4.a.a 1 20.e even 4 1
32.4.a.c yes 1 5.c odd 4 1
64.4.a.a 1 40.i odd 4 1
64.4.a.e 1 40.k even 4 1
256.4.b.c 2 80.i odd 4 1
256.4.b.c 2 80.t odd 4 1
256.4.b.e 2 80.j even 4 1
256.4.b.e 2 80.s even 4 1
288.4.a.h 1 60.l odd 4 1
288.4.a.i 1 15.e even 4 1
576.4.a.g 1 120.q odd 4 1
576.4.a.h 1 120.w even 4 1
800.4.a.a 1 5.c odd 4 1
800.4.a.k 1 20.e even 4 1
800.4.c.a 2 1.a even 1 1 trivial
800.4.c.a 2 5.b even 2 1 inner
800.4.c.b 2 4.b odd 2 1
800.4.c.b 2 20.d odd 2 1
1568.4.a.c 1 35.f even 4 1
1568.4.a.o 1 140.j odd 4 1
1600.4.a.e 1 40.k even 4 1
1600.4.a.bw 1 40.i odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(800, [\chi])$$:

 $$T_{3}^{2} + 64$$ T3^2 + 64 $$T_{11} + 40$$ T11 + 40

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 64$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 256$$
$11$ $$(T + 40)^{2}$$
$13$ $$T^{2} + 2500$$
$17$ $$T^{2} + 900$$
$19$ $$(T + 40)^{2}$$
$23$ $$T^{2} + 2304$$
$29$ $$(T - 34)^{2}$$
$31$ $$(T - 320)^{2}$$
$37$ $$T^{2} + 96100$$
$41$ $$(T - 410)^{2}$$
$43$ $$T^{2} + 23104$$
$47$ $$T^{2} + 173056$$
$53$ $$T^{2} + 168100$$
$59$ $$(T - 200)^{2}$$
$61$ $$(T - 30)^{2}$$
$67$ $$T^{2} + 602176$$
$71$ $$(T - 400)^{2}$$
$73$ $$T^{2} + 396900$$
$79$ $$(T - 1120)^{2}$$
$83$ $$T^{2} + 304704$$
$89$ $$(T - 326)^{2}$$
$97$ $$T^{2} + 12100$$