Properties

Label 6001.12
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([286,329]))
 
pari: [g,chi] = znchar(Mod(12,6001))
 

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.ep

\(\chi_{6001}(12,\cdot)\) \(\chi_{6001}(20,\cdot)\) \(\chi_{6001}(28,\cdot)\) \(\chi_{6001}(129,\cdot)\) \(\chi_{6001}(142,\cdot)\) \(\chi_{6001}(214,\cdot)\) \(\chi_{6001}(215,\cdot)\) \(\chi_{6001}(228,\cdot)\) \(\chi_{6001}(279,\cdot)\) \(\chi_{6001}(299,\cdot)\) \(\chi_{6001}(301,\cdot)\) \(\chi_{6001}(367,\cdot)\) \(\chi_{6001}(377,\cdot)\) \(\chi_{6001}(380,\cdot)\) \(\chi_{6001}(465,\cdot)\) \(\chi_{6001}(496,\cdot)\) \(\chi_{6001}(498,\cdot)\) \(\chi_{6001}(532,\cdot)\) \(\chi_{6001}(573,\cdot)\) \(\chi_{6001}(626,\cdot)\) \(\chi_{6001}(643,\cdot)\) \(\chi_{6001}(651,\cdot)\) \(\chi_{6001}(711,\cdot)\) \(\chi_{6001}(719,\cdot)\) \(\chi_{6001}(775,\cdot)\) \(\chi_{6001}(811,\cdot)\) \(\chi_{6001}(830,\cdot)\) \(\chi_{6001}(844,\cdot)\) \(\chi_{6001}(949,\cdot)\) \(\chi_{6001}(955,\cdot)\) ...

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

\((2825,3180)\) → \((e\left(\frac{13}{16}\right),e\left(\frac{329}{352}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(1\)\(1\)\(e\left(\frac{29}{44}\right)\)\(e\left(\frac{263}{352}\right)\)\(e\left(\frac{7}{22}\right)\)\(e\left(\frac{307}{352}\right)\)\(e\left(\frac{13}{32}\right)\)\(e\left(\frac{15}{32}\right)\)\(e\left(\frac{43}{44}\right)\)\(e\left(\frac{87}{176}\right)\)\(e\left(\frac{17}{32}\right)\)\(e\left(\frac{135}{176}\right)\)
value at e.g. 2

Related number fields

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