Properties

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

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[61]
 
pari: [g,chi] = znchar(Mod(61,4000))
 

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.dd
Orbit index = 82

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 ]
 

\(\chi_{4000}(21,\cdot)\) \(\chi_{4000}(61,\cdot)\) \(\chi_{4000}(141,\cdot)\) \(\chi_{4000}(181,\cdot)\) \(\chi_{4000}(221,\cdot)\) \(\chi_{4000}(261,\cdot)\) \(\chi_{4000}(341,\cdot)\) \(\chi_{4000}(381,\cdot)\) \(\chi_{4000}(421,\cdot)\) \(\chi_{4000}(461,\cdot)\) \(\chi_{4000}(541,\cdot)\) \(\chi_{4000}(581,\cdot)\) \(\chi_{4000}(621,\cdot)\) \(\chi_{4000}(661,\cdot)\) \(\chi_{4000}(741,\cdot)\) \(\chi_{4000}(781,\cdot)\) \(\chi_{4000}(821,\cdot)\) \(\chi_{4000}(861,\cdot)\) \(\chi_{4000}(941,\cdot)\) \(\chi_{4000}(981,\cdot)\) \(\chi_{4000}(1021,\cdot)\) \(\chi_{4000}(1061,\cdot)\) \(\chi_{4000}(1141,\cdot)\) \(\chi_{4000}(1181,\cdot)\) \(\chi_{4000}(1221,\cdot)\) \(\chi_{4000}(1261,\cdot)\) \(\chi_{4000}(1341,\cdot)\) \(\chi_{4000}(1381,\cdot)\) \(\chi_{4000}(1421,\cdot)\) \(\chi_{4000}(1461,\cdot)\) ...

Values on generators

\((2751,2501,1377)\) → \((1,e\left(\frac{3}{8}\right),e\left(\frac{14}{25}\right))\)

Values

-1137911131719212327
\(1\)\(1\)\(e\left(\frac{9}{200}\right)\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{9}{100}\right)\)\(e\left(\frac{87}{200}\right)\)\(e\left(\frac{93}{200}\right)\)\(e\left(\frac{19}{50}\right)\)\(e\left(\frac{141}{200}\right)\)\(e\left(\frac{79}{200}\right)\)\(e\left(\frac{61}{100}\right)\)\(e\left(\frac{27}{200}\right)\)
value at  e.g. 2

Related number fields

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