Properties

Conductor 1337
Order 57
Real no
Primitive yes
Minimal yes
Parity even
Orbit label 1337.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(1337)
 
sage: chi = H[345]
 
pari: [g,chi] = znchar(Mod(345,1337))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 1337
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 57
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 = even
Orbit label = 1337.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 ]
 

\(\chi_{1337}(25,\cdot)\) \(\chi_{1337}(30,\cdot)\) \(\chi_{1337}(32,\cdot)\) \(\chi_{1337}(107,\cdot)\) \(\chi_{1337}(121,\cdot)\) \(\chi_{1337}(177,\cdot)\) \(\chi_{1337}(221,\cdot)\) \(\chi_{1337}(298,\cdot)\) \(\chi_{1337}(312,\cdot)\) \(\chi_{1337}(345,\cdot)\) \(\chi_{1337}(368,\cdot)\) \(\chi_{1337}(387,\cdot)\) \(\chi_{1337}(536,\cdot)\) \(\chi_{1337}(562,\cdot)\) \(\chi_{1337}(578,\cdot)\) \(\chi_{1337}(625,\cdot)\) \(\chi_{1337}(709,\cdot)\) \(\chi_{1337}(723,\cdot)\) \(\chi_{1337}(753,\cdot)\) \(\chi_{1337}(800,\cdot)\) \(\chi_{1337}(816,\cdot)\) \(\chi_{1337}(900,\cdot)\) \(\chi_{1337}(914,\cdot)\) \(\chi_{1337}(961,\cdot)\) \(\chi_{1337}(991,\cdot)\) \(\chi_{1337}(1024,\cdot)\) \(\chi_{1337}(1080,\cdot)\) \(\chi_{1337}(1108,\cdot)\) \(\chi_{1337}(1115,\cdot)\) \(\chi_{1337}(1152,\cdot)\) ...

Values on generators

\((192,974)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{11}{19}\right))\)

Values

-112345689101112
\(1\)\(1\)\(e\left(\frac{8}{57}\right)\)\(e\left(\frac{28}{57}\right)\)\(e\left(\frac{16}{57}\right)\)\(e\left(\frac{35}{57}\right)\)\(e\left(\frac{12}{19}\right)\)\(e\left(\frac{8}{19}\right)\)\(e\left(\frac{56}{57}\right)\)\(e\left(\frac{43}{57}\right)\)\(e\left(\frac{31}{57}\right)\)\(e\left(\frac{44}{57}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{57})\)