Properties

Label 6001.363
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([30,11]))
 
pari: [g,chi] = znchar(Mod(363,6001))
 

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.by

\(\chi_{6001}(363,\cdot)\) \(\chi_{6001}(490,\cdot)\) \(\chi_{6001}(2051,\cdot)\) \(\chi_{6001}(2370,\cdot)\) \(\chi_{6001}(3040,\cdot)\) \(\chi_{6001}(3184,\cdot)\) \(\chi_{6001}(3777,\cdot)\) \(\chi_{6001}(4177,\cdot)\) \(\chi_{6001}(4579,\cdot)\) \(\chi_{6001}(4583,\cdot)\) \(\chi_{6001}(4695,\cdot)\) \(\chi_{6001}(4935,\cdot)\) \(\chi_{6001}(5009,\cdot)\) \(\chi_{6001}(5396,\cdot)\) \(\chi_{6001}(5654,\cdot)\) \(\chi_{6001}(5707,\cdot)\)

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

\((2825,3180)\) → \((e\left(\frac{15}{16}\right),e\left(\frac{11}{32}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(1\)\(1\)\(-1\)\(e\left(\frac{9}{32}\right)\)\(1\)\(e\left(\frac{29}{32}\right)\)\(e\left(\frac{25}{32}\right)\)\(e\left(\frac{11}{32}\right)\)\(-1\)\(e\left(\frac{9}{16}\right)\)\(e\left(\frac{13}{32}\right)\)\(e\left(\frac{3}{16}\right)\)
value at e.g. 2

Related number fields

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