Properties

Conductor 4003
Order 2001
Real No
Primitive Yes
Parity Even
Orbit Label 4003.o

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(4003)
 
sage: chi = H[4]
 
pari: [g,chi] = znchar(Mod(4,4003))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 4003
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 2001
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 = 4003.o
Orbit index = 15

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 ]
 

\(\chi_{4003}(4,\cdot)\) \(\chi_{4003}(7,\cdot)\) \(\chi_{4003}(9,\cdot)\) \(\chi_{4003}(16,\cdot)\) \(\chi_{4003}(19,\cdot)\) \(\chi_{4003}(24,\cdot)\) \(\chi_{4003}(25,\cdot)\) \(\chi_{4003}(29,\cdot)\) \(\chi_{4003}(37,\cdot)\) \(\chi_{4003}(40,\cdot)\) \(\chi_{4003}(42,\cdot)\) \(\chi_{4003}(44,\cdot)\) \(\chi_{4003}(46,\cdot)\) \(\chi_{4003}(49,\cdot)\) \(\chi_{4003}(52,\cdot)\) \(\chi_{4003}(53,\cdot)\) \(\chi_{4003}(54,\cdot)\) \(\chi_{4003}(59,\cdot)\) \(\chi_{4003}(60,\cdot)\) \(\chi_{4003}(62,\cdot)\) \(\chi_{4003}(63,\cdot)\) \(\chi_{4003}(67,\cdot)\) \(\chi_{4003}(68,\cdot)\) \(\chi_{4003}(77,\cdot)\) \(\chi_{4003}(79,\cdot)\) \(\chi_{4003}(81,\cdot)\) \(\chi_{4003}(82,\cdot)\) \(\chi_{4003}(83,\cdot)\) \(\chi_{4003}(89,\cdot)\) \(\chi_{4003}(91,\cdot)\) ...

Values on generators

\(2\) → \(e\left(\frac{1}{2001}\right)\)

Values

-11234567891011
\(1\)\(1\)\(e\left(\frac{1}{2001}\right)\)\(e\left(\frac{416}{2001}\right)\)\(e\left(\frac{2}{2001}\right)\)\(e\left(\frac{1471}{2001}\right)\)\(e\left(\frac{139}{667}\right)\)\(e\left(\frac{133}{2001}\right)\)\(e\left(\frac{1}{667}\right)\)\(e\left(\frac{832}{2001}\right)\)\(e\left(\frac{64}{87}\right)\)\(e\left(\frac{562}{667}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{2001})\)