Properties

Conductor 3997
Order 570
Real no
Primitive yes
Minimal yes
Parity odd
Orbit label 3997.cz

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
 
sage: H = DirichletGroup_conrey(3997)
 
sage: chi = H[692]
 
pari: [g,chi] = znchar(Mod(692,3997))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 3997
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 570
Real = no
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
 
Primitive = yes
Minimal = yes
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 
Parity = odd
Orbit label = 3997.cz
Orbit index = 78

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 ]
 

\(\chi_{3997}(13,\cdot)\) \(\chi_{3997}(34,\cdot)\) \(\chi_{3997}(83,\cdot)\) \(\chi_{3997}(97,\cdot)\) \(\chi_{3997}(139,\cdot)\) \(\chi_{3997}(174,\cdot)\) \(\chi_{3997}(202,\cdot)\) \(\chi_{3997}(223,\cdot)\) \(\chi_{3997}(237,\cdot)\) \(\chi_{3997}(244,\cdot)\) \(\chi_{3997}(279,\cdot)\) \(\chi_{3997}(293,\cdot)\) \(\chi_{3997}(321,\cdot)\) \(\chi_{3997}(342,\cdot)\) \(\chi_{3997}(482,\cdot)\) \(\chi_{3997}(531,\cdot)\) \(\chi_{3997}(538,\cdot)\) \(\chi_{3997}(552,\cdot)\) \(\chi_{3997}(559,\cdot)\) \(\chi_{3997}(580,\cdot)\) \(\chi_{3997}(601,\cdot)\) \(\chi_{3997}(608,\cdot)\) \(\chi_{3997}(615,\cdot)\) \(\chi_{3997}(671,\cdot)\) \(\chi_{3997}(692,\cdot)\) \(\chi_{3997}(762,\cdot)\) \(\chi_{3997}(790,\cdot)\) \(\chi_{3997}(804,\cdot)\) \(\chi_{3997}(832,\cdot)\) \(\chi_{3997}(846,\cdot)\) ...

Values on generators

\((2285,1716)\) → \((-1,e\left(\frac{268}{285}\right))\)

Values

-112345689101112
\(-1\)\(1\)\(e\left(\frac{28}{57}\right)\)\(e\left(\frac{251}{570}\right)\)\(e\left(\frac{56}{57}\right)\)\(e\left(\frac{187}{570}\right)\)\(e\left(\frac{177}{190}\right)\)\(e\left(\frac{9}{19}\right)\)\(e\left(\frac{251}{285}\right)\)\(e\left(\frac{467}{570}\right)\)\(e\left(\frac{4}{285}\right)\)\(e\left(\frac{241}{570}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{285})\)