Properties

Label 6001.185
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([33,16]))
 
pari: [g,chi] = znchar(Mod(185,6001))
 

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.db

\(\chi_{6001}(185,\cdot)\) \(\chi_{6001}(484,\cdot)\) \(\chi_{6001}(570,\cdot)\) \(\chi_{6001}(893,\cdot)\) \(\chi_{6001}(937,\cdot)\) \(\chi_{6001}(1199,\cdot)\) \(\chi_{6001}(1290,\cdot)\) \(\chi_{6001}(1396,\cdot)\) \(\chi_{6001}(1470,\cdot)\) \(\chi_{6001}(1668,\cdot)\) \(\chi_{6001}(1749,\cdot)\) \(\chi_{6001}(1787,\cdot)\) \(\chi_{6001}(1896,\cdot)\) \(\chi_{6001}(2021,\cdot)\) \(\chi_{6001}(2140,\cdot)\) \(\chi_{6001}(2303,\cdot)\) \(\chi_{6001}(2688,\cdot)\) \(\chi_{6001}(2882,\cdot)\) \(\chi_{6001}(3011,\cdot)\) \(\chi_{6001}(3041,\cdot)\) \(\chi_{6001}(3317,\cdot)\) \(\chi_{6001}(3364,\cdot)\) \(\chi_{6001}(3408,\cdot)\) \(\chi_{6001}(3670,\cdot)\) \(\chi_{6001}(3715,\cdot)\) \(\chi_{6001}(3867,\cdot)\) \(\chi_{6001}(4014,\cdot)\) \(\chi_{6001}(4139,\cdot)\) \(\chi_{6001}(4258,\cdot)\) \(\chi_{6001}(4367,\cdot)\) ...

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

\((2825,3180)\) → \((e\left(\frac{3}{8}\right),e\left(\frac{2}{11}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(1\)\(1\)\(e\left(\frac{3}{44}\right)\)\(e\left(\frac{49}{88}\right)\)\(e\left(\frac{3}{22}\right)\)\(e\left(\frac{5}{88}\right)\)\(e\left(\frac{5}{8}\right)\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{9}{44}\right)\)\(e\left(\frac{5}{44}\right)\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{47}{88}\right)\)
value at e.g. 2

Related number fields

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