Properties

Conductor 1000
Order 100
Real No
Primitive Yes
Parity Odd
Orbit Label 1000.bj

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(1000)
 
sage: chi = H[477]
 
pari: [g,chi] = znchar(Mod(477,1000))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 1000
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 100
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 = Odd
Orbit label = 1000.bj
Orbit index = 36

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 ]
 

\(\chi_{1000}(13,\cdot)\) \(\chi_{1000}(37,\cdot)\) \(\chi_{1000}(53,\cdot)\) \(\chi_{1000}(77,\cdot)\) \(\chi_{1000}(117,\cdot)\) \(\chi_{1000}(133,\cdot)\) \(\chi_{1000}(173,\cdot)\) \(\chi_{1000}(197,\cdot)\) \(\chi_{1000}(213,\cdot)\) \(\chi_{1000}(237,\cdot)\) \(\chi_{1000}(253,\cdot)\) \(\chi_{1000}(277,\cdot)\) \(\chi_{1000}(317,\cdot)\) \(\chi_{1000}(333,\cdot)\) \(\chi_{1000}(373,\cdot)\) \(\chi_{1000}(397,\cdot)\) \(\chi_{1000}(413,\cdot)\) \(\chi_{1000}(437,\cdot)\) \(\chi_{1000}(453,\cdot)\) \(\chi_{1000}(477,\cdot)\) \(\chi_{1000}(517,\cdot)\) \(\chi_{1000}(533,\cdot)\) \(\chi_{1000}(573,\cdot)\) \(\chi_{1000}(597,\cdot)\) \(\chi_{1000}(613,\cdot)\) \(\chi_{1000}(637,\cdot)\) \(\chi_{1000}(653,\cdot)\) \(\chi_{1000}(677,\cdot)\) \(\chi_{1000}(717,\cdot)\) \(\chi_{1000}(733,\cdot)\) ...

Values on generators

\((751,501,377)\) → \((1,-1,e\left(\frac{81}{100}\right))\)

Values

-1137911131719212327
\(-1\)\(1\)\(e\left(\frac{17}{100}\right)\)\(e\left(\frac{17}{20}\right)\)\(e\left(\frac{17}{50}\right)\)\(e\left(\frac{3}{50}\right)\)\(e\left(\frac{9}{100}\right)\)\(e\left(\frac{13}{100}\right)\)\(e\left(\frac{2}{25}\right)\)\(e\left(\frac{1}{50}\right)\)\(e\left(\frac{11}{100}\right)\)\(e\left(\frac{51}{100}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{100})\)