Properties

Conductor 1000
Order 50
Real No
Primitive No
Parity Odd
Orbit Label 4000.ci

Related objects

Learn more about

Show commands for: SageMath / Pari/GP
sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
sage: H = DirichletGroup_conrey(4000)
sage: chi = H[111]
pari: [g,chi] = znchar(Mod(111,4000))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 1000
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 50
Real = No
sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
Primitive = No
sage: chi.is_odd()
pari: zncharisodd(g,chi)
Parity = Odd
Orbit label = 4000.ci
Orbit index = 61

Galois orbit

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

\(\chi_{4000}(111,\cdot)\) \(\chi_{4000}(271,\cdot)\) \(\chi_{4000}(431,\cdot)\) \(\chi_{4000}(591,\cdot)\) \(\chi_{4000}(911,\cdot)\) \(\chi_{4000}(1071,\cdot)\) \(\chi_{4000}(1231,\cdot)\) \(\chi_{4000}(1391,\cdot)\) \(\chi_{4000}(1711,\cdot)\) \(\chi_{4000}(1871,\cdot)\) \(\chi_{4000}(2031,\cdot)\) \(\chi_{4000}(2191,\cdot)\) \(\chi_{4000}(2511,\cdot)\) \(\chi_{4000}(2671,\cdot)\) \(\chi_{4000}(2831,\cdot)\) \(\chi_{4000}(2991,\cdot)\) \(\chi_{4000}(3311,\cdot)\) \(\chi_{4000}(3471,\cdot)\) \(\chi_{4000}(3631,\cdot)\) \(\chi_{4000}(3791,\cdot)\)

Inducing primitive character

\(\chi_{1000}(611,\cdot)\)

Values on generators

\((2751,2501,1377)\) → \((-1,-1,e\left(\frac{9}{25}\right))\)

Values

-1137911131719212327
\(-1\)\(1\)\(e\left(\frac{13}{25}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{1}{25}\right)\)\(e\left(\frac{9}{25}\right)\)\(e\left(\frac{27}{50}\right)\)\(e\left(\frac{7}{25}\right)\)\(e\left(\frac{12}{25}\right)\)\(e\left(\frac{31}{50}\right)\)\(e\left(\frac{33}{50}\right)\)\(e\left(\frac{14}{25}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{25})\)