# Properties

 Label 800.2.c.g Level $800$ Weight $2$ Character orbit 800.c Analytic conductor $6.388$ 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] = [800,2,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, 2, names="a")

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

chi := DirichletCharacter("800.449");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$800 = 2^{5} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ 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: $$6.38803216170$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 3x^{2} + 1$$ x^4 + 3*x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{3} - 2 \beta_{2} q^{7} - 2 q^{9}+O(q^{10})$$ q + b2 * q^3 - 2*b2 * q^7 - 2 * q^9 $$q + \beta_{2} q^{3} - 2 \beta_{2} q^{7} - 2 q^{9} + \beta_{3} q^{11} + 4 \beta_1 q^{13} + 7 \beta_1 q^{17} + 3 \beta_{3} q^{19} + 10 q^{21} - 2 \beta_{2} q^{23} + \beta_{2} q^{27} + 2 \beta_{3} q^{31} + 5 \beta_1 q^{33} - 2 \beta_1 q^{37} - 4 \beta_{3} q^{39} + 5 q^{41} + 4 \beta_{2} q^{47} - 13 q^{49} - 7 \beta_{3} q^{51} + 6 \beta_1 q^{53} + 15 \beta_1 q^{57} - 4 \beta_{3} q^{59} + 10 q^{61} + 4 \beta_{2} q^{63} + \beta_{2} q^{67} + 10 q^{69} - 4 \beta_{3} q^{71} - 9 \beta_1 q^{73} - 10 \beta_1 q^{77} + 2 \beta_{3} q^{79} - 11 q^{81} - 5 \beta_{2} q^{83} + 5 q^{89} + 8 \beta_{3} q^{91} + 10 \beta_1 q^{93} - 2 \beta_1 q^{97} - 2 \beta_{3} q^{99}+O(q^{100})$$ q + b2 * q^3 - 2*b2 * q^7 - 2 * q^9 + b3 * q^11 + 4*b1 * q^13 + 7*b1 * q^17 + 3*b3 * q^19 + 10 * q^21 - 2*b2 * q^23 + b2 * q^27 + 2*b3 * q^31 + 5*b1 * q^33 - 2*b1 * q^37 - 4*b3 * q^39 + 5 * q^41 + 4*b2 * q^47 - 13 * q^49 - 7*b3 * q^51 + 6*b1 * q^53 + 15*b1 * q^57 - 4*b3 * q^59 + 10 * q^61 + 4*b2 * q^63 + b2 * q^67 + 10 * q^69 - 4*b3 * q^71 - 9*b1 * q^73 - 10*b1 * q^77 + 2*b3 * q^79 - 11 * q^81 - 5*b2 * q^83 + 5 * q^89 + 8*b3 * q^91 + 10*b1 * q^93 - 2*b1 * q^97 - 2*b3 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 8 q^{9}+O(q^{10})$$ 4 * q - 8 * q^9 $$4 q - 8 q^{9} + 40 q^{21} + 20 q^{41} - 52 q^{49} + 40 q^{61} + 40 q^{69} - 44 q^{81} + 20 q^{89}+O(q^{100})$$ 4 * q - 8 * q^9 + 40 * q^21 + 20 * q^41 - 52 * q^49 + 40 * q^61 + 40 * q^69 - 44 * q^81 + 20 * q^89

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 3x^{2} + 1$$ :

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

## 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
 − 1.61803i − 0.618034i 1.61803i 0.618034i
0 2.23607i 0 0 0 4.47214i 0 −2.00000 0
449.2 0 2.23607i 0 0 0 4.47214i 0 −2.00000 0
449.3 0 2.23607i 0 0 0 4.47214i 0 −2.00000 0
449.4 0 2.23607i 0 0 0 4.47214i 0 −2.00000 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
5.b even 2 1 inner
20.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 800.2.c.g 4
3.b odd 2 1 7200.2.f.bg 4
4.b odd 2 1 inner 800.2.c.g 4
5.b even 2 1 inner 800.2.c.g 4
5.c odd 4 1 800.2.a.k 2
5.c odd 4 1 800.2.a.l yes 2
8.b even 2 1 1600.2.c.o 4
8.d odd 2 1 1600.2.c.o 4
12.b even 2 1 7200.2.f.bg 4
15.d odd 2 1 7200.2.f.bg 4
15.e even 4 1 7200.2.a.cf 2
15.e even 4 1 7200.2.a.cn 2
20.d odd 2 1 inner 800.2.c.g 4
20.e even 4 1 800.2.a.k 2
20.e even 4 1 800.2.a.l yes 2
40.e odd 2 1 1600.2.c.o 4
40.f even 2 1 1600.2.c.o 4
40.i odd 4 1 1600.2.a.ba 2
40.i odd 4 1 1600.2.a.bb 2
40.k even 4 1 1600.2.a.ba 2
40.k even 4 1 1600.2.a.bb 2
60.h even 2 1 7200.2.f.bg 4
60.l odd 4 1 7200.2.a.cf 2
60.l odd 4 1 7200.2.a.cn 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
800.2.a.k 2 5.c odd 4 1
800.2.a.k 2 20.e even 4 1
800.2.a.l yes 2 5.c odd 4 1
800.2.a.l yes 2 20.e even 4 1
800.2.c.g 4 1.a even 1 1 trivial
800.2.c.g 4 4.b odd 2 1 inner
800.2.c.g 4 5.b even 2 1 inner
800.2.c.g 4 20.d odd 2 1 inner
1600.2.a.ba 2 40.i odd 4 1
1600.2.a.ba 2 40.k even 4 1
1600.2.a.bb 2 40.i odd 4 1
1600.2.a.bb 2 40.k even 4 1
1600.2.c.o 4 8.b even 2 1
1600.2.c.o 4 8.d odd 2 1
1600.2.c.o 4 40.e odd 2 1
1600.2.c.o 4 40.f even 2 1
7200.2.a.cf 2 15.e even 4 1
7200.2.a.cf 2 60.l odd 4 1
7200.2.a.cn 2 15.e even 4 1
7200.2.a.cn 2 60.l odd 4 1
7200.2.f.bg 4 3.b odd 2 1
7200.2.f.bg 4 12.b even 2 1
7200.2.f.bg 4 15.d odd 2 1
7200.2.f.bg 4 60.h even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(800, [\chi])$$:

 $$T_{3}^{2} + 5$$ T3^2 + 5 $$T_{11}^{2} - 5$$ T11^2 - 5

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T^{2} + 5)^{2}$$
$5$ $$T^{4}$$
$7$ $$(T^{2} + 20)^{2}$$
$11$ $$(T^{2} - 5)^{2}$$
$13$ $$(T^{2} + 16)^{2}$$
$17$ $$(T^{2} + 49)^{2}$$
$19$ $$(T^{2} - 45)^{2}$$
$23$ $$(T^{2} + 20)^{2}$$
$29$ $$T^{4}$$
$31$ $$(T^{2} - 20)^{2}$$
$37$ $$(T^{2} + 4)^{2}$$
$41$ $$(T - 5)^{4}$$
$43$ $$T^{4}$$
$47$ $$(T^{2} + 80)^{2}$$
$53$ $$(T^{2} + 36)^{2}$$
$59$ $$(T^{2} - 80)^{2}$$
$61$ $$(T - 10)^{4}$$
$67$ $$(T^{2} + 5)^{2}$$
$71$ $$(T^{2} - 80)^{2}$$
$73$ $$(T^{2} + 81)^{2}$$
$79$ $$(T^{2} - 20)^{2}$$
$83$ $$(T^{2} + 125)^{2}$$
$89$ $$(T - 5)^{4}$$
$97$ $$(T^{2} + 4)^{2}$$