Properties

Modulus 1849
Conductor 43
Order 7
Real no
Primitive no
Minimal no
Parity even
Orbit label 1849.e

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(1849)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([3]))
 
pari: [g,chi] = znchar(Mod(78,1849))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Modulus = 1849
Conductor = 43
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 7
Real = no
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
 
Primitive = no
Minimal = no
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 
Parity = even
Orbit label = 1849.e
Orbit index = 5

Galois orbit

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

\(\chi_{1849}(78,\cdot)\) \(\chi_{1849}(403,\cdot)\) \(\chi_{1849}(537,\cdot)\) \(\chi_{1849}(1208,\cdot)\) \(\chi_{1849}(1546,\cdot)\) \(\chi_{1849}(1774,\cdot)\)

Values on generators

\(3\) → \(e\left(\frac{3}{7}\right)\)

Values

-11234567891011
\(1\)\(1\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{5}{7}\right)\)\(1\)\(1\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{6}{7}\right)\)
value at  e.g. 2

Related number fields

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