Properties

Label 6001.83
Modulus $6001$
Conductor $6001$
Order $88$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(6001)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([33,75]))
 
pari: [g,chi] = znchar(Mod(83,6001))
 

Basic properties

Modulus: \(6001\)
Conductor: \(6001\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(88\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6001.dd

\(\chi_{6001}(83,\cdot)\) \(\chi_{6001}(434,\cdot)\) \(\chi_{6001}(695,\cdot)\) \(\chi_{6001}(723,\cdot)\) \(\chi_{6001}(882,\cdot)\) \(\chi_{6001}(1080,\cdot)\) \(\chi_{6001}(1120,\cdot)\) \(\chi_{6001}(1328,\cdot)\) \(\chi_{6001}(1368,\cdot)\) \(\chi_{6001}(1606,\cdot)\) \(\chi_{6001}(1692,\cdot)\) \(\chi_{6001}(2089,\cdot)\) \(\chi_{6001}(2110,\cdot)\) \(\chi_{6001}(2116,\cdot)\) \(\chi_{6001}(2150,\cdot)\) \(\chi_{6001}(2246,\cdot)\) \(\chi_{6001}(2582,\cdot)\) \(\chi_{6001}(2756,\cdot)\) \(\chi_{6001}(2915,\cdot)\) \(\chi_{6001}(2933,\cdot)\) \(\chi_{6001}(3068,\cdot)\) \(\chi_{6001}(3086,\cdot)\) \(\chi_{6001}(3245,\cdot)\) \(\chi_{6001}(3419,\cdot)\) \(\chi_{6001}(3755,\cdot)\) \(\chi_{6001}(3851,\cdot)\) \(\chi_{6001}(3885,\cdot)\) \(\chi_{6001}(3891,\cdot)\) \(\chi_{6001}(3912,\cdot)\) \(\chi_{6001}(4309,\cdot)\) ...

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Values on generators

\((2825,3180)\) → \((e\left(\frac{3}{8}\right),e\left(\frac{75}{88}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(1\)\(1\)\(e\left(\frac{1}{44}\right)\)\(e\left(\frac{5}{22}\right)\)\(e\left(\frac{1}{22}\right)\)\(e\left(\frac{5}{22}\right)\)\(i\)\(-1\)\(e\left(\frac{3}{44}\right)\)\(e\left(\frac{5}{11}\right)\)\(i\)\(e\left(\frac{67}{88}\right)\)
value at e.g. 2

Related number fields

Field of values: $\Q(\zeta_{88})$
Fixed field: Number field defined by a degree 88 polynomial