Properties

Conductor 8041
Order 8
Real No
Primitive Yes
Parity Even
Orbit Label 8041.z

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(8041)
 
sage: chi = H[2837]
 
pari: [g,chi] = znchar(Mod(2837,8041))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 8041
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 8
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 = 8041.z
Orbit index = 26

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 ]
 

\(\chi_{8041}(2837,\cdot)\) \(\chi_{8041}(3783,\cdot)\) \(\chi_{8041}(6621,\cdot)\) \(\chi_{8041}(7567,\cdot)\)

Values on generators

\((6580,2366,562)\) → \((-1,e\left(\frac{3}{8}\right),-1)\)

Values

-11234567891012
\(1\)\(1\)\(i\)\(e\left(\frac{7}{8}\right)\)\(-1\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{1}{8}\right)\)\(-i\)\(-i\)\(e\left(\frac{5}{8}\right)\)\(e\left(\frac{3}{8}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{8})\)