Properties

Conductor 4001
Order 2000
Real No
Primitive Yes
Parity Even
Orbit Label 4001.w

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(4001)
 
sage: chi = H[9]
 
pari: [g,chi] = znchar(Mod(9,4001))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 4001
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 2000
Real = No
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
 
Primitive = Yes
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 
Parity = Even
Orbit label = 4001.w
Orbit index = 23

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 ]
 

\(\chi_{4001}(9,\cdot)\) \(\chi_{4001}(18,\cdot)\) \(\chi_{4001}(29,\cdot)\) \(\chi_{4001}(31,\cdot)\) \(\chi_{4001}(33,\cdot)\) \(\chi_{4001}(36,\cdot)\) \(\chi_{4001}(45,\cdot)\) \(\chi_{4001}(47,\cdot)\) \(\chi_{4001}(61,\cdot)\) \(\chi_{4001}(62,\cdot)\) \(\chi_{4001}(66,\cdot)\) \(\chi_{4001}(72,\cdot)\) \(\chi_{4001}(83,\cdot)\) \(\chi_{4001}(89,\cdot)\) \(\chi_{4001}(90,\cdot)\) \(\chi_{4001}(94,\cdot)\) \(\chi_{4001}(97,\cdot)\) \(\chi_{4001}(111,\cdot)\) \(\chi_{4001}(116,\cdot)\) \(\chi_{4001}(117,\cdot)\) \(\chi_{4001}(118,\cdot)\) \(\chi_{4001}(122,\cdot)\) \(\chi_{4001}(124,\cdot)\) \(\chi_{4001}(126,\cdot)\) \(\chi_{4001}(137,\cdot)\) \(\chi_{4001}(138,\cdot)\) \(\chi_{4001}(145,\cdot)\) \(\chi_{4001}(149,\cdot)\) \(\chi_{4001}(155,\cdot)\) \(\chi_{4001}(165,\cdot)\) ...

Values on generators

\(3\) → \(e\left(\frac{1}{2000}\right)\)

Values

-11234567891011
\(1\)\(1\)\(e\left(\frac{23}{500}\right)\)\(e\left(\frac{1}{2000}\right)\)\(e\left(\frac{23}{250}\right)\)\(e\left(\frac{59}{100}\right)\)\(e\left(\frac{93}{2000}\right)\)\(e\left(\frac{57}{500}\right)\)\(e\left(\frac{69}{500}\right)\)\(e\left(\frac{1}{1000}\right)\)\(e\left(\frac{159}{250}\right)\)\(e\left(\frac{57}{80}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{2000})\)