Properties

Conductor 731
Order 42
Real No
Primitive No
Parity Even
Orbit Label 8041.cx

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[67]
 
pari: [g,chi] = znchar(Mod(67,8041))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 731
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 42
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 = 8041.cx
Orbit index = 76

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 ]
 

\(\chi_{8041}(67,\cdot)\) \(\chi_{8041}(1002,\cdot)\) \(\chi_{8041}(1563,\cdot)\) \(\chi_{8041}(2124,\cdot)\) \(\chi_{8041}(3807,\cdot)\) \(\chi_{8041}(3994,\cdot)\) \(\chi_{8041}(4181,\cdot)\) \(\chi_{8041}(4368,\cdot)\) \(\chi_{8041}(4555,\cdot)\) \(\chi_{8041}(5303,\cdot)\) \(\chi_{8041}(6051,\cdot)\) \(\chi_{8041}(7921,\cdot)\)

Inducing primitive character

\(\chi_{731}(67,\cdot)\)

Values on generators

\((6580,2366,562)\) → \((1,-1,e\left(\frac{20}{21}\right))\)

Values

-11234567891012
\(1\)\(1\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{19}{42}\right)\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{13}{42}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{19}{21}\right)\)\(e\left(\frac{1}{42}\right)\)\(e\left(\frac{37}{42}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{21})\)