Properties

Conductor 4000
Order 200
Real no
Primitive yes
Minimal yes
Parity odd
Orbit label 4000.de

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[37]
 
pari: [g,chi] = znchar(Mod(37,4000))
 

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 = odd
Orbit label = 4000.de
Orbit index = 83

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 ]
 

\(\chi_{4000}(13,\cdot)\) \(\chi_{4000}(37,\cdot)\) \(\chi_{4000}(117,\cdot)\) \(\chi_{4000}(173,\cdot)\) \(\chi_{4000}(197,\cdot)\) \(\chi_{4000}(253,\cdot)\) \(\chi_{4000}(277,\cdot)\) \(\chi_{4000}(333,\cdot)\) \(\chi_{4000}(413,\cdot)\) \(\chi_{4000}(437,\cdot)\) \(\chi_{4000}(517,\cdot)\) \(\chi_{4000}(573,\cdot)\) \(\chi_{4000}(597,\cdot)\) \(\chi_{4000}(653,\cdot)\) \(\chi_{4000}(677,\cdot)\) \(\chi_{4000}(733,\cdot)\) \(\chi_{4000}(813,\cdot)\) \(\chi_{4000}(837,\cdot)\) \(\chi_{4000}(917,\cdot)\) \(\chi_{4000}(973,\cdot)\) \(\chi_{4000}(997,\cdot)\) \(\chi_{4000}(1053,\cdot)\) \(\chi_{4000}(1077,\cdot)\) \(\chi_{4000}(1133,\cdot)\) \(\chi_{4000}(1213,\cdot)\) \(\chi_{4000}(1237,\cdot)\) \(\chi_{4000}(1317,\cdot)\) \(\chi_{4000}(1373,\cdot)\) \(\chi_{4000}(1397,\cdot)\) \(\chi_{4000}(1453,\cdot)\) ...

Values on generators

\((2751,2501,1377)\) → \((1,e\left(\frac{1}{8}\right),e\left(\frac{29}{100}\right))\)

Values

-1137911131719212327
\(-1\)\(1\)\(e\left(\frac{81}{200}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{81}{100}\right)\)\(e\left(\frac{133}{200}\right)\)\(e\left(\frac{37}{200}\right)\)\(e\left(\frac{67}{100}\right)\)\(e\left(\frac{19}{200}\right)\)\(e\left(\frac{61}{200}\right)\)\(e\left(\frac{37}{50}\right)\)\(e\left(\frac{43}{200}\right)\)
value at  e.g. 2

Related number fields

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