Properties

Conductor 2003
Order 26
Real No
Primitive Yes
Parity Odd
Orbit Label 2003.h

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(2003)
 
sage: chi = H[45]
 
pari: [g,chi] = znchar(Mod(45,2003))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 2003
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 26
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 = Odd
Orbit label = 2003.h
Orbit index = 8

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 ]
 

\(\chi_{2003}(45,\cdot)\) \(\chi_{2003}(87,\cdot)\) \(\chi_{2003}(91,\cdot)\) \(\chi_{2003}(95,\cdot)\) \(\chi_{2003}(443,\cdot)\) \(\chi_{2003}(990,\cdot)\) \(\chi_{2003}(1370,\cdot)\) \(\chi_{2003}(1519,\cdot)\) \(\chi_{2003}(1734,\cdot)\) \(\chi_{2003}(1750,\cdot)\) \(\chi_{2003}(1914,\cdot)\) \(\chi_{2003}(1981,\cdot)\)

Values on generators

\(5\) → \(e\left(\frac{9}{26}\right)\)

Values

-11234567891011
\(-1\)\(1\)\(e\left(\frac{21}{26}\right)\)\(e\left(\frac{10}{13}\right)\)\(e\left(\frac{8}{13}\right)\)\(e\left(\frac{9}{26}\right)\)\(e\left(\frac{15}{26}\right)\)\(e\left(\frac{15}{26}\right)\)\(e\left(\frac{11}{26}\right)\)\(e\left(\frac{7}{13}\right)\)\(e\left(\frac{2}{13}\right)\)\(e\left(\frac{25}{26}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{13})\)