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[517]
pari: [g,chi] = znchar(Mod(517,1000))

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

\((751,501,377)\) → \((1,-1,e\left(\frac{73}{100}\right))\)

Values

-1137911131719212327
\(-1\)\(1\)\(e\left(\frac{61}{100}\right)\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{11}{50}\right)\)\(e\left(\frac{49}{50}\right)\)\(e\left(\frac{97}{100}\right)\)\(e\left(\frac{29}{100}\right)\)\(e\left(\frac{16}{25}\right)\)\(e\left(\frac{33}{50}\right)\)\(e\left(\frac{63}{100}\right)\)\(e\left(\frac{83}{100}\right)\)
value at  e.g. 2

Related number fields

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