Properties

Label 400.27
Modulus 400400
Conductor 400400
Order 2020
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(400, base_ring=CyclotomicField(20))
 
M = H._module
 
chi = DirichletCharacter(H, M([10,5,1]))
 
pari: [g,chi] = znchar(Mod(27,400))
 

Basic properties

Modulus: 400400
Conductor: 400400
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: 2020
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 400.bd

χ400(3,)\chi_{400}(3,\cdot) χ400(27,)\chi_{400}(27,\cdot) χ400(83,)\chi_{400}(83,\cdot) χ400(163,)\chi_{400}(163,\cdot) χ400(187,)\chi_{400}(187,\cdot) χ400(267,)\chi_{400}(267,\cdot) χ400(323,)\chi_{400}(323,\cdot) χ400(347,)\chi_{400}(347,\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(ζ20)\Q(\zeta_{20})
Fixed field: 20.20.104857600000000000000000000000000000000000.2

Values on generators

(351,101,177)(351,101,177)(1,i,e(120))(-1,i,e\left(\frac{1}{20}\right))

First values

aa 1-1113377991111131317171919212123232727
χ400(27,a) \chi_{ 400 }(27, a) 1111e(35)e\left(\frac{3}{5}\right)iie(15)e\left(\frac{1}{5}\right)e(1120)e\left(\frac{11}{20}\right)e(710)e\left(\frac{7}{10}\right)e(1320)e\left(\frac{13}{20}\right)e(320)e\left(\frac{3}{20}\right)e(1720)e\left(\frac{17}{20}\right)e(1120)e\left(\frac{11}{20}\right)e(45)e\left(\frac{4}{5}\right)
sage: chi.jacobi_sum(n)
 
χ400(27,a)   \chi_{ 400 }(27,a) \; at   a=\;a = e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
τa(χ400(27,))   \tau_{ a }( \chi_{ 400 }(27,·) )\; at   a=\;a = e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
J(χ400(27,),χ400(n,))   J(\chi_{ 400 }(27,·),\chi_{ 400 }(n,·)) \; for   n= \; n = e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
K(a,b,χ400(27,))  K(a,b,\chi_{ 400 }(27,·)) \; at   a,b=\; a,b = e.g. 1,2