Properties

Label 6001.111
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([11,27]))
 
pari: [g,chi] = znchar(Mod(111,6001))
 

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.cx

\(\chi_{6001}(111,\cdot)\) \(\chi_{6001}(321,\cdot)\) \(\chi_{6001}(355,\cdot)\) \(\chi_{6001}(382,\cdot)\) \(\chi_{6001}(444,\cdot)\) \(\chi_{6001}(597,\cdot)\) \(\chi_{6001}(774,\cdot)\) \(\chi_{6001}(865,\cdot)\) \(\chi_{6001}(1103,\cdot)\) \(\chi_{6001}(1284,\cdot)\) \(\chi_{6001}(1351,\cdot)\) \(\chi_{6001}(1420,\cdot)\) \(\chi_{6001}(1589,\cdot)\) \(\chi_{6001}(1776,\cdot)\) \(\chi_{6001}(1838,\cdot)\) \(\chi_{6001}(2202,\cdot)\) \(\chi_{6001}(2388,\cdot)\) \(\chi_{6001}(2450,\cdot)\) \(\chi_{6001}(2807,\cdot)\) \(\chi_{6001}(2905,\cdot)\) \(\chi_{6001}(3096,\cdot)\) \(\chi_{6001}(3194,\cdot)\) \(\chi_{6001}(3551,\cdot)\) \(\chi_{6001}(3613,\cdot)\) \(\chi_{6001}(3799,\cdot)\) \(\chi_{6001}(4163,\cdot)\) \(\chi_{6001}(4225,\cdot)\) \(\chi_{6001}(4412,\cdot)\) \(\chi_{6001}(4581,\cdot)\) \(\chi_{6001}(4650,\cdot)\) ...

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

\((2825,3180)\) → \((e\left(\frac{1}{8}\right),e\left(\frac{27}{88}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(1\)\(1\)\(e\left(\frac{3}{44}\right)\)\(e\left(\frac{19}{44}\right)\)\(e\left(\frac{3}{22}\right)\)\(e\left(\frac{19}{44}\right)\)\(-1\)\(-i\)\(e\left(\frac{9}{44}\right)\)\(e\left(\frac{19}{22}\right)\)\(-1\)\(e\left(\frac{25}{88}\right)\)
value at e.g. 2

Related number fields

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