Properties

Label 6001.56
Modulus $6001$
Conductor $6001$
Order $352$
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([110,239]))
 
pari: [g,chi] = znchar(Mod(56,6001))
 

Basic properties

Modulus: \(6001\)
Conductor: \(6001\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(352\)
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.ee

\(\chi_{6001}(56,\cdot)\) \(\chi_{6001}(96,\cdot)\) \(\chi_{6001}(210,\cdot)\) \(\chi_{6001}(241,\cdot)\) \(\chi_{6001}(248,\cdot)\) \(\chi_{6001}(284,\cdot)\) \(\chi_{6001}(313,\cdot)\) \(\chi_{6001}(326,\cdot)\) \(\chi_{6001}(329,\cdot)\) \(\chi_{6001}(415,\cdot)\) \(\chi_{6001}(430,\cdot)\) \(\chi_{6001}(486,\cdot)\) \(\chi_{6001}(487,\cdot)\) \(\chi_{6001}(564,\cdot)\) \(\chi_{6001}(572,\cdot)\) \(\chi_{6001}(583,\cdot)\) \(\chi_{6001}(592,\cdot)\) \(\chi_{6001}(686,\cdot)\) \(\chi_{6001}(692,\cdot)\) \(\chi_{6001}(737,\cdot)\) \(\chi_{6001}(758,\cdot)\) \(\chi_{6001}(760,\cdot)\) \(\chi_{6001}(772,\cdot)\) \(\chi_{6001}(785,\cdot)\) \(\chi_{6001}(796,\cdot)\) \(\chi_{6001}(845,\cdot)\) \(\chi_{6001}(847,\cdot)\) \(\chi_{6001}(857,\cdot)\) \(\chi_{6001}(908,\cdot)\) \(\chi_{6001}(930,\cdot)\) ...

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

\((2825,3180)\) → \((e\left(\frac{5}{16}\right),e\left(\frac{239}{352}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(1\)\(1\)\(e\left(\frac{8}{11}\right)\)\(e\left(\frac{349}{352}\right)\)\(e\left(\frac{5}{11}\right)\)\(e\left(\frac{129}{352}\right)\)\(e\left(\frac{23}{32}\right)\)\(e\left(\frac{21}{32}\right)\)\(e\left(\frac{2}{11}\right)\)\(e\left(\frac{173}{176}\right)\)\(e\left(\frac{3}{32}\right)\)\(e\left(\frac{163}{176}\right)\)
value at e.g. 2

Related number fields

Field of values: $\Q(\zeta_{352})$
Fixed field: Number field defined by a degree 352 polynomial