Properties

Label 6001.42
Modulus $6001$
Conductor $6001$
Order $8$
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([5,6]))
 
pari: [g,chi] = znchar(Mod(42,6001))
 

Basic properties

Modulus: \(6001\)
Conductor: \(6001\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(8\)
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.q

\(\chi_{6001}(42,\cdot)\) \(\chi_{6001}(1454,\cdot)\) \(\chi_{6001}(2076,\cdot)\) \(\chi_{6001}(2429,\cdot)\)

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

\((2825,3180)\) → \((e\left(\frac{5}{8}\right),-i)\)

Values

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

Related number fields

Field of values: \(\Q(\zeta_{8})\)
Fixed field: 8.8.793945482553449180612017.4