Properties

Label 6001.2
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([77,41]))
 
pari: [g,chi] = znchar(Mod(2,6001))
 

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.dc

\(\chi_{6001}(2,\cdot)\) \(\chi_{6001}(8,\cdot)\) \(\chi_{6001}(32,\cdot)\) \(\chi_{6001}(128,\cdot)\) \(\chi_{6001}(512,\cdot)\) \(\chi_{6001}(638,\cdot)\) \(\chi_{6001}(750,\cdot)\) \(\chi_{6001}(790,\cdot)\) \(\chi_{6001}(797,\cdot)\) \(\chi_{6001}(950,\cdot)\) \(\chi_{6001}(1301,\cdot)\) \(\chi_{6001}(1794,\cdot)\) \(\chi_{6001}(2191,\cdot)\) \(\chi_{6001}(2201,\cdot)\) \(\chi_{6001}(2552,\cdot)\) \(\chi_{6001}(2763,\cdot)\) \(\chi_{6001}(2803,\cdot)\) \(\chi_{6001}(2813,\cdot)\) \(\chi_{6001}(2841,\cdot)\) \(\chi_{6001}(3000,\cdot)\) \(\chi_{6001}(3001,\cdot)\) \(\chi_{6001}(3160,\cdot)\) \(\chi_{6001}(3188,\cdot)\) \(\chi_{6001}(3198,\cdot)\) \(\chi_{6001}(3238,\cdot)\) \(\chi_{6001}(3449,\cdot)\) \(\chi_{6001}(3800,\cdot)\) \(\chi_{6001}(3810,\cdot)\) \(\chi_{6001}(4207,\cdot)\) \(\chi_{6001}(4700,\cdot)\) ...

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

\((2825,3180)\) → \((e\left(\frac{7}{8}\right),e\left(\frac{41}{88}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(1\)\(1\)\(e\left(\frac{29}{44}\right)\)\(e\left(\frac{15}{44}\right)\)\(e\left(\frac{7}{22}\right)\)\(e\left(\frac{15}{44}\right)\)\(1\)\(-i\)\(e\left(\frac{43}{44}\right)\)\(e\left(\frac{15}{22}\right)\)\(1\)\(e\left(\frac{7}{88}\right)\)
value at e.g. 2

Related number fields

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