Properties

Conductor 4001
Order 2000
Real No
Primitive No
Parity Even
Orbit Label 8002.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(8002)
 
sage: chi = H[33]
 
pari: [g,chi] = znchar(Mod(33,8002))
 

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 = No
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 
Parity = Even
Orbit label = 8002.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 ]
 

\(\chi_{8002}(9,\cdot)\) \(\chi_{8002}(29,\cdot)\) \(\chi_{8002}(31,\cdot)\) \(\chi_{8002}(33,\cdot)\) \(\chi_{8002}(45,\cdot)\) \(\chi_{8002}(47,\cdot)\) \(\chi_{8002}(61,\cdot)\) \(\chi_{8002}(83,\cdot)\) \(\chi_{8002}(89,\cdot)\) \(\chi_{8002}(97,\cdot)\) \(\chi_{8002}(111,\cdot)\) \(\chi_{8002}(117,\cdot)\) \(\chi_{8002}(137,\cdot)\) \(\chi_{8002}(145,\cdot)\) \(\chi_{8002}(149,\cdot)\) \(\chi_{8002}(155,\cdot)\) \(\chi_{8002}(165,\cdot)\) \(\chi_{8002}(171,\cdot)\) \(\chi_{8002}(197,\cdot)\) \(\chi_{8002}(203,\cdot)\) \(\chi_{8002}(225,\cdot)\) \(\chi_{8002}(231,\cdot)\) \(\chi_{8002}(235,\cdot)\) \(\chi_{8002}(253,\cdot)\) \(\chi_{8002}(257,\cdot)\) \(\chi_{8002}(269,\cdot)\) \(\chi_{8002}(289,\cdot)\) \(\chi_{8002}(305,\cdot)\) \(\chi_{8002}(339,\cdot)\) \(\chi_{8002}(377,\cdot)\) ...

Inducing primitive character

\(\chi_{4001}(33,\cdot)\)

Values on generators

\(3\) → \(e\left(\frac{713}{2000}\right)\)

Values

-113579111315171921
\(1\)\(1\)\(e\left(\frac{713}{2000}\right)\)\(e\left(\frac{67}{100}\right)\)\(e\left(\frac{141}{500}\right)\)\(e\left(\frac{713}{1000}\right)\)\(e\left(\frac{1}{80}\right)\)\(e\left(\frac{233}{500}\right)\)\(e\left(\frac{53}{2000}\right)\)\(e\left(\frac{303}{2000}\right)\)\(e\left(\frac{453}{500}\right)\)\(e\left(\frac{1277}{2000}\right)\)
value at  e.g. 2

Related number fields

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