Properties

Label 6001.50
Modulus $6001$
Conductor $6001$
Order $176$
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([88,39]))
 
pari: [g,chi] = znchar(Mod(50,6001))
 

Basic properties

Modulus: \(6001\)
Conductor: \(6001\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(176\)
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.dz

\(\chi_{6001}(50,\cdot)\) \(\chi_{6001}(152,\cdot)\) \(\chi_{6001}(169,\cdot)\) \(\chi_{6001}(186,\cdot)\) \(\chi_{6001}(254,\cdot)\) \(\chi_{6001}(271,\cdot)\) \(\chi_{6001}(288,\cdot)\) \(\chi_{6001}(509,\cdot)\) \(\chi_{6001}(628,\cdot)\) \(\chi_{6001}(747,\cdot)\) \(\chi_{6001}(798,\cdot)\) \(\chi_{6001}(1036,\cdot)\) \(\chi_{6001}(1172,\cdot)\) \(\chi_{6001}(1189,\cdot)\) \(\chi_{6001}(1223,\cdot)\) \(\chi_{6001}(1240,\cdot)\) \(\chi_{6001}(1257,\cdot)\) \(\chi_{6001}(1393,\cdot)\) \(\chi_{6001}(1427,\cdot)\) \(\chi_{6001}(1750,\cdot)\) \(\chi_{6001}(1784,\cdot)\) \(\chi_{6001}(1920,\cdot)\) \(\chi_{6001}(1937,\cdot)\) \(\chi_{6001}(1954,\cdot)\) \(\chi_{6001}(1988,\cdot)\) \(\chi_{6001}(2005,\cdot)\) \(\chi_{6001}(2141,\cdot)\) \(\chi_{6001}(2379,\cdot)\) \(\chi_{6001}(2430,\cdot)\) \(\chi_{6001}(2549,\cdot)\) ...

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

\((2825,3180)\) → \((-1,e\left(\frac{39}{176}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(1\)\(1\)\(e\left(\frac{15}{44}\right)\)\(e\left(\frac{127}{176}\right)\)\(e\left(\frac{15}{22}\right)\)\(e\left(\frac{171}{176}\right)\)\(e\left(\frac{1}{16}\right)\)\(e\left(\frac{7}{16}\right)\)\(e\left(\frac{1}{44}\right)\)\(e\left(\frac{39}{88}\right)\)\(e\left(\frac{5}{16}\right)\)\(e\left(\frac{35}{44}\right)\)
value at e.g. 2

Related number fields

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