Properties

Conductor 4000
Order 200
Real no
Primitive yes
Minimal yes
Parity even
Orbit label 4000.cy

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[3]
 
pari: [g,chi] = znchar(Mod(3,4000))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 4000
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 200
Real = no
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
 
Primitive = yes
Minimal = yes
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 
Parity = even
Orbit label = 4000.cy
Orbit index = 77

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 ]
 

\(\chi_{4000}(3,\cdot)\) \(\chi_{4000}(27,\cdot)\) \(\chi_{4000}(83,\cdot)\) \(\chi_{4000}(163,\cdot)\) \(\chi_{4000}(187,\cdot)\) \(\chi_{4000}(267,\cdot)\) \(\chi_{4000}(323,\cdot)\) \(\chi_{4000}(347,\cdot)\) \(\chi_{4000}(403,\cdot)\) \(\chi_{4000}(427,\cdot)\) \(\chi_{4000}(483,\cdot)\) \(\chi_{4000}(563,\cdot)\) \(\chi_{4000}(587,\cdot)\) \(\chi_{4000}(667,\cdot)\) \(\chi_{4000}(723,\cdot)\) \(\chi_{4000}(747,\cdot)\) \(\chi_{4000}(803,\cdot)\) \(\chi_{4000}(827,\cdot)\) \(\chi_{4000}(883,\cdot)\) \(\chi_{4000}(963,\cdot)\) \(\chi_{4000}(987,\cdot)\) \(\chi_{4000}(1067,\cdot)\) \(\chi_{4000}(1123,\cdot)\) \(\chi_{4000}(1147,\cdot)\) \(\chi_{4000}(1203,\cdot)\) \(\chi_{4000}(1227,\cdot)\) \(\chi_{4000}(1283,\cdot)\) \(\chi_{4000}(1363,\cdot)\) \(\chi_{4000}(1387,\cdot)\) \(\chi_{4000}(1467,\cdot)\) ...

Values on generators

\((2751,2501,1377)\) → \((-1,e\left(\frac{3}{8}\right),e\left(\frac{7}{100}\right))\)

Values

-1137911131719212327
\(1\)\(1\)\(e\left(\frac{23}{200}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{23}{100}\right)\)\(e\left(\frac{139}{200}\right)\)\(e\left(\frac{71}{200}\right)\)\(e\left(\frac{61}{100}\right)\)\(e\left(\frac{77}{200}\right)\)\(e\left(\frac{63}{200}\right)\)\(e\left(\frac{23}{25}\right)\)\(e\left(\frac{69}{200}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{200})\)