Properties

Label 720.109
Modulus 720720
Conductor 8080
Order 44
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(720, base_ring=CyclotomicField(4))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,3,0,2]))
 
pari: [g,chi] = znchar(Mod(109,720))
 

Basic properties

Modulus: 720720
Conductor: 8080
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: 44
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from χ80(29,)\chi_{80}(29,\cdot)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 720.bm

χ720(109,)\chi_{720}(109,\cdot) χ720(469,)\chi_{720}(469,\cdot)

sage: chi.galois_orbit()
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: Q(i)\mathbb{Q}(i)
Fixed field: 4.4.51200.1

Values on generators

(271,181,641,577)(271,181,641,577)(1,i,1,1)(1,-i,1,-1)

First values

aa 1-11177111113131717191923232929313137374141
χ720(109,a) \chi_{ 720 }(109, a) 111111i-ii-i1-1ii11ii11ii1-1
sage: chi.jacobi_sum(n)
 
χ720(109,a)   \chi_{ 720 }(109,a) \; at   a=\;a = e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
τa(χ720(109,))   \tau_{ a }( \chi_{ 720 }(109,·) )\; at   a=\;a = e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
J(χ720(109,),χ720(n,))   J(\chi_{ 720 }(109,·),\chi_{ 720 }(n,·)) \; for   n= \; n = e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
K(a,b,χ720(109,))  K(a,b,\chi_{ 720 }(109,·)) \; at   a,b=\; a,b = e.g. 1,2