Properties

Label 2000.123
Modulus 20002000
Conductor 20002000
Order 100100
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2000, base_ring=CyclotomicField(100))
 
M = H._module
 
chi = DirichletCharacter(H, M([50,25,51]))
 
pari: [g,chi] = znchar(Mod(123,2000))
 

Basic properties

Modulus: 20002000
Conductor: 20002000
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: 100100
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 2000.by

χ2000(67,)\chi_{2000}(67,\cdot) χ2000(123,)\chi_{2000}(123,\cdot) χ2000(147,)\chi_{2000}(147,\cdot) χ2000(203,)\chi_{2000}(203,\cdot) χ2000(227,)\chi_{2000}(227,\cdot) χ2000(283,)\chi_{2000}(283,\cdot) χ2000(363,)\chi_{2000}(363,\cdot) χ2000(387,)\chi_{2000}(387,\cdot) χ2000(467,)\chi_{2000}(467,\cdot) χ2000(523,)\chi_{2000}(523,\cdot) χ2000(547,)\chi_{2000}(547,\cdot) χ2000(603,)\chi_{2000}(603,\cdot) χ2000(627,)\chi_{2000}(627,\cdot) χ2000(683,)\chi_{2000}(683,\cdot) χ2000(763,)\chi_{2000}(763,\cdot) χ2000(787,)\chi_{2000}(787,\cdot) χ2000(867,)\chi_{2000}(867,\cdot) χ2000(923,)\chi_{2000}(923,\cdot) χ2000(947,)\chi_{2000}(947,\cdot) χ2000(1003,)\chi_{2000}(1003,\cdot) χ2000(1027,)\chi_{2000}(1027,\cdot) χ2000(1083,)\chi_{2000}(1083,\cdot) χ2000(1163,)\chi_{2000}(1163,\cdot) χ2000(1187,)\chi_{2000}(1187,\cdot) χ2000(1267,)\chi_{2000}(1267,\cdot) χ2000(1323,)\chi_{2000}(1323,\cdot) χ2000(1347,)\chi_{2000}(1347,\cdot) χ2000(1403,)\chi_{2000}(1403,\cdot) χ2000(1427,)\chi_{2000}(1427,\cdot) χ2000(1483,)\chi_{2000}(1483,\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(ζ100)\Q(\zeta_{100})
Fixed field: Number field defined by a degree 100 polynomial

Values on generators

(751,501,1377)(751,501,1377)(1,i,e(51100))(-1,i,e\left(\frac{51}{100}\right))

First values

aa 1-1113377991111131317171919212123232727
χ2000(123,a) \chi_{ 2000 }(123, a) 1111e(4150)e\left(\frac{41}{50}\right)e(720)e\left(\frac{7}{20}\right)e(1625)e\left(\frac{16}{25}\right)e(51100)e\left(\frac{51}{100}\right)e(1625)e\left(\frac{16}{25}\right)e(23100)e\left(\frac{23}{100}\right)e(43100)e\left(\frac{43}{100}\right)e(17100)e\left(\frac{17}{100}\right)e(81100)e\left(\frac{81}{100}\right)e(2350)e\left(\frac{23}{50}\right)
sage: chi.jacobi_sum(n)
 
χ2000(123,a)   \chi_{ 2000 }(123,a) \; at   a=\;a = e.g. 2