Properties

Label 6001.18
Modulus $6001$
Conductor $353$
Order $176$
Real no
Primitive no
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([0,83]))
 
pari: [g,chi] = znchar(Mod(18,6001))
 

Basic properties

Modulus: \(6001\)
Conductor: \(353\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(176\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{353}(18,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6001.dy

\(\chi_{6001}(18,\cdot)\) \(\chi_{6001}(86,\cdot)\) \(\chi_{6001}(120,\cdot)\) \(\chi_{6001}(188,\cdot)\) \(\chi_{6001}(307,\cdot)\) \(\chi_{6001}(392,\cdot)\) \(\chi_{6001}(562,\cdot)\) \(\chi_{6001}(579,\cdot)\) \(\chi_{6001}(613,\cdot)\) \(\chi_{6001}(630,\cdot)\) \(\chi_{6001}(681,\cdot)\) \(\chi_{6001}(715,\cdot)\) \(\chi_{6001}(749,\cdot)\) \(\chi_{6001}(800,\cdot)\) \(\chi_{6001}(902,\cdot)\) \(\chi_{6001}(987,\cdot)\) \(\chi_{6001}(1021,\cdot)\) \(\chi_{6001}(1089,\cdot)\) \(\chi_{6001}(1106,\cdot)\) \(\chi_{6001}(1157,\cdot)\) \(\chi_{6001}(1259,\cdot)\) \(\chi_{6001}(1565,\cdot)\) \(\chi_{6001}(1667,\cdot)\) \(\chi_{6001}(1718,\cdot)\) \(\chi_{6001}(1735,\cdot)\) \(\chi_{6001}(1803,\cdot)\) \(\chi_{6001}(1837,\cdot)\) \(\chi_{6001}(1922,\cdot)\) \(\chi_{6001}(2024,\cdot)\) \(\chi_{6001}(2075,\cdot)\) ...

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

\((2825,3180)\) → \((1,e\left(\frac{83}{176}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(1\)\(1\)\(e\left(\frac{15}{44}\right)\)\(e\left(\frac{83}{176}\right)\)\(e\left(\frac{15}{22}\right)\)\(e\left(\frac{127}{176}\right)\)\(e\left(\frac{13}{16}\right)\)\(e\left(\frac{11}{16}\right)\)\(e\left(\frac{1}{44}\right)\)\(e\left(\frac{83}{88}\right)\)\(e\left(\frac{1}{16}\right)\)\(e\left(\frac{13}{44}\right)\)
value at e.g. 2

Related number fields

Field of values: $\Q(\zeta_{176})$
Fixed field: Number field defined by a degree 176 polynomial