Properties

Conductor 2009
Order 14
Real No
Primitive Yes
Parity Even
Orbit Label 2009.u

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(2009)
 
sage: chi = H[778]
 
pari: [g,chi] = znchar(Mod(778,2009))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 2009
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 = Yes
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 
Parity = Even
Orbit label = 2009.u
Orbit index = 21

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 ]
 

\(\chi_{2009}(204,\cdot)\) \(\chi_{2009}(778,\cdot)\) \(\chi_{2009}(1065,\cdot)\) \(\chi_{2009}(1352,\cdot)\) \(\chi_{2009}(1639,\cdot)\) \(\chi_{2009}(1926,\cdot)\)

Values on generators

\((493,785)\) → \((e\left(\frac{1}{7}\right),-1)\)

Values

-112345689101112
\(1\)\(1\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{9}{14}\right)\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{5}{14}\right)\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{3}{14}\right)\)\(e\left(\frac{1}{14}\right)\)
value at  e.g. 2

Related number fields

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