Properties

Conductor 731
Order 14
Real No
Primitive No
Parity Even
Orbit Label 8041.bm

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[441]
 
pari: [g,chi] = znchar(Mod(441,8041))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 731
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 14
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.bm
Orbit index = 39

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 ]
 

\(\chi_{8041}(441,\cdot)\) \(\chi_{8041}(815,\cdot)\) \(\chi_{8041}(2498,\cdot)\) \(\chi_{8041}(3246,\cdot)\) \(\chi_{8041}(5864,\cdot)\) \(\chi_{8041}(7173,\cdot)\)

Inducing primitive character

\(\chi_{731}(441,\cdot)\)

Values on generators

\((6580,2366,562)\) → \((1,-1,e\left(\frac{5}{7}\right))\)

Values

-11234567891012
\(1\)\(1\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{3}{14}\right)\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{5}{14}\right)\)\(-1\)\(-1\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{9}{14}\right)\)\(e\left(\frac{11}{14}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{7})\)