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Properties

Label 100.j
Modulus 100100
Conductor 100100
Order 1010
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(100, base_ring=CyclotomicField(10))
 
M = H._module
 
chi = DirichletCharacter(H, M([5,8]))
 
chi.galois_orbit()
 
[g,chi] = znchar(Mod(11,100))
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: 100100
Conductor: 100100
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: 1010
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Related number fields

Field of values: Q(ζ5)\Q(\zeta_{5})
Fixed field: 10.0.156250000000000.1

Characters in Galois orbit

Character −1-1 11 33 77 99 1111 1313 1717 1919 2121 2323 2727
χ100(11,⋅)\chi_{100}(11,\cdot) −1-1 11 e(110)e\left(\frac{1}{10}\right) −1-1 e(15)e\left(\frac{1}{5}\right) e(310)e\left(\frac{3}{10}\right) e(15)e\left(\frac{1}{5}\right) e(25)e\left(\frac{2}{5}\right) e(910)e\left(\frac{9}{10}\right) e(35)e\left(\frac{3}{5}\right) e(310)e\left(\frac{3}{10}\right) e(310)e\left(\frac{3}{10}\right)
χ100(31,⋅)\chi_{100}(31,\cdot) −1-1 11 e(310)e\left(\frac{3}{10}\right) −1-1 e(35)e\left(\frac{3}{5}\right) e(910)e\left(\frac{9}{10}\right) e(35)e\left(\frac{3}{5}\right) e(15)e\left(\frac{1}{5}\right) e(710)e\left(\frac{7}{10}\right) e(45)e\left(\frac{4}{5}\right) e(910)e\left(\frac{9}{10}\right) e(910)e\left(\frac{9}{10}\right)
χ100(71,⋅)\chi_{100}(71,\cdot) −1-1 11 e(710)e\left(\frac{7}{10}\right) −1-1 e(25)e\left(\frac{2}{5}\right) e(110)e\left(\frac{1}{10}\right) e(25)e\left(\frac{2}{5}\right) e(45)e\left(\frac{4}{5}\right) e(310)e\left(\frac{3}{10}\right) e(15)e\left(\frac{1}{5}\right) e(110)e\left(\frac{1}{10}\right) e(110)e\left(\frac{1}{10}\right)
χ100(91,⋅)\chi_{100}(91,\cdot) −1-1 11 e(910)e\left(\frac{9}{10}\right) −1-1 e(45)e\left(\frac{4}{5}\right) e(710)e\left(\frac{7}{10}\right) e(45)e\left(\frac{4}{5}\right) e(35)e\left(\frac{3}{5}\right) e(110)e\left(\frac{1}{10}\right) e(25)e\left(\frac{2}{5}\right) e(710)e\left(\frac{7}{10}\right) e(710)e\left(\frac{7}{10}\right)