Properties

Label 6001.252
Modulus $6001$
Conductor $6001$
Order $32$
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([18,7]))
 
pari: [g,chi] = znchar(Mod(252,6001))
 

Basic properties

Modulus: \(6001\)
Conductor: \(6001\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(32\)
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.cb

\(\chi_{6001}(252,\cdot)\) \(\chi_{6001}(420,\cdot)\) \(\chi_{6001}(700,\cdot)\) \(\chi_{6001}(843,\cdot)\) \(\chi_{6001}(1405,\cdot)\) \(\chi_{6001}(1706,\cdot)\) \(\chi_{6001}(1771,\cdot)\) \(\chi_{6001}(3167,\cdot)\) \(\chi_{6001}(3236,\cdot)\) \(\chi_{6001}(3278,\cdot)\) \(\chi_{6001}(3393,\cdot)\) \(\chi_{6001}(3424,\cdot)\) \(\chi_{6001}(3463,\cdot)\) \(\chi_{6001}(4342,\cdot)\) \(\chi_{6001}(4952,\cdot)\) \(\chi_{6001}(5655,\cdot)\)

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

\((2825,3180)\) → \((e\left(\frac{9}{16}\right),e\left(\frac{7}{32}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(1\)\(1\)\(-i\)\(e\left(\frac{25}{32}\right)\)\(-1\)\(e\left(\frac{13}{32}\right)\)\(e\left(\frac{17}{32}\right)\)\(e\left(\frac{27}{32}\right)\)\(i\)\(e\left(\frac{9}{16}\right)\)\(e\left(\frac{5}{32}\right)\)\(e\left(\frac{1}{16}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{32})\)
Fixed field: Number field defined by a degree 32 polynomial