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Properties

Label 725.x
Modulus 725725
Conductor 725725
Order 2020
Real no
Primitive yes
Minimal yes
Parity even

Related objects

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(725, base_ring=CyclotomicField(20))
 
M = H._module
 
chi = DirichletCharacter(H, M([9,5]))
 
chi.galois_orbit()
 
[g,chi] = znchar(Mod(12,725))
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: 725725
Conductor: 725725
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)
 

Related number fields

Field of values: Q(ζ20)\Q(\zeta_{20})
Fixed field: Number field defined by a degree 20 polynomial

Characters in Galois orbit

Character −1-1 11 22 33 44 66 77 88 99 1111 1212 1313
χ725(12,⋅)\chi_{725}(12,\cdot) 11 11 e(710)e\left(\frac{7}{10}\right) e(25)e\left(\frac{2}{5}\right) e(25)e\left(\frac{2}{5}\right) e(110)e\left(\frac{1}{10}\right) ii e(110)e\left(\frac{1}{10}\right) e(45)e\left(\frac{4}{5}\right) e(920)e\left(\frac{9}{20}\right) e(45)e\left(\frac{4}{5}\right) e(120)e\left(\frac{1}{20}\right)
χ725(133,⋅)\chi_{725}(133,\cdot) 11 11 e(910)e\left(\frac{9}{10}\right) e(45)e\left(\frac{4}{5}\right) e(45)e\left(\frac{4}{5}\right) e(710)e\left(\frac{7}{10}\right) −i-i e(710)e\left(\frac{7}{10}\right) e(35)e\left(\frac{3}{5}\right) e(320)e\left(\frac{3}{20}\right) e(35)e\left(\frac{3}{5}\right) e(720)e\left(\frac{7}{20}\right)
χ725(278,⋅)\chi_{725}(278,\cdot) 11 11 e(110)e\left(\frac{1}{10}\right) e(15)e\left(\frac{1}{5}\right) e(15)e\left(\frac{1}{5}\right) e(310)e\left(\frac{3}{10}\right) −i-i e(310)e\left(\frac{3}{10}\right) e(25)e\left(\frac{2}{5}\right) e(720)e\left(\frac{7}{20}\right) e(25)e\left(\frac{2}{5}\right) e(320)e\left(\frac{3}{20}\right)
χ725(302,⋅)\chi_{725}(302,\cdot) 11 11 e(310)e\left(\frac{3}{10}\right) e(35)e\left(\frac{3}{5}\right) e(35)e\left(\frac{3}{5}\right) e(910)e\left(\frac{9}{10}\right) ii e(910)e\left(\frac{9}{10}\right) e(15)e\left(\frac{1}{5}\right) e(120)e\left(\frac{1}{20}\right) e(15)e\left(\frac{1}{5}\right) e(920)e\left(\frac{9}{20}\right)
χ725(423,⋅)\chi_{725}(423,\cdot) 11 11 e(310)e\left(\frac{3}{10}\right) e(35)e\left(\frac{3}{5}\right) e(35)e\left(\frac{3}{5}\right) e(910)e\left(\frac{9}{10}\right) −i-i e(910)e\left(\frac{9}{10}\right) e(15)e\left(\frac{1}{5}\right) e(1120)e\left(\frac{11}{20}\right) e(15)e\left(\frac{1}{5}\right) e(1920)e\left(\frac{19}{20}\right)
χ725(447,⋅)\chi_{725}(447,\cdot) 11 11 e(110)e\left(\frac{1}{10}\right) e(15)e\left(\frac{1}{5}\right) e(15)e\left(\frac{1}{5}\right) e(310)e\left(\frac{3}{10}\right) ii e(310)e\left(\frac{3}{10}\right) e(25)e\left(\frac{2}{5}\right) e(1720)e\left(\frac{17}{20}\right) e(25)e\left(\frac{2}{5}\right) e(1320)e\left(\frac{13}{20}\right)
χ725(592,⋅)\chi_{725}(592,\cdot) 11 11 e(910)e\left(\frac{9}{10}\right) e(45)e\left(\frac{4}{5}\right) e(45)e\left(\frac{4}{5}\right) e(710)e\left(\frac{7}{10}\right) ii e(710)e\left(\frac{7}{10}\right) e(35)e\left(\frac{3}{5}\right) e(1320)e\left(\frac{13}{20}\right) e(35)e\left(\frac{3}{5}\right) e(1720)e\left(\frac{17}{20}\right)
χ725(713,⋅)\chi_{725}(713,\cdot) 11 11 e(710)e\left(\frac{7}{10}\right) e(25)e\left(\frac{2}{5}\right) e(25)e\left(\frac{2}{5}\right) e(110)e\left(\frac{1}{10}\right) −i-i e(110)e\left(\frac{1}{10}\right) e(45)e\left(\frac{4}{5}\right) e(1920)e\left(\frac{19}{20}\right) e(45)e\left(\frac{4}{5}\right) e(1120)e\left(\frac{11}{20}\right)