Properties

Label 6001.3
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([22,1]))
 
pari: [g,chi] = znchar(Mod(3,6001))
 

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.er

\(\chi_{6001}(3,\cdot)\) \(\chi_{6001}(5,\cdot)\) \(\chi_{6001}(27,\cdot)\) \(\chi_{6001}(45,\cdot)\) \(\chi_{6001}(74,\cdot)\) \(\chi_{6001}(75,\cdot)\) \(\chi_{6001}(112,\cdot)\) \(\chi_{6001}(124,\cdot)\) \(\chi_{6001}(125,\cdot)\) \(\chi_{6001}(133,\cdot)\) \(\chi_{6001}(147,\cdot)\) \(\chi_{6001}(243,\cdot)\) \(\chi_{6001}(245,\cdot)\) \(\chi_{6001}(379,\cdot)\) \(\chi_{6001}(381,\cdot)\) \(\chi_{6001}(386,\cdot)\) \(\chi_{6001}(405,\cdot)\) \(\chi_{6001}(516,\cdot)\) \(\chi_{6001}(533,\cdot)\) \(\chi_{6001}(568,\cdot)\) \(\chi_{6001}(635,\cdot)\) \(\chi_{6001}(666,\cdot)\) \(\chi_{6001}(675,\cdot)\) \(\chi_{6001}(694,\cdot)\) \(\chi_{6001}(726,\cdot)\) \(\chi_{6001}(768,\cdot)\) \(\chi_{6001}(809,\cdot)\) \(\chi_{6001}(838,\cdot)\) \(\chi_{6001}(856,\cdot)\) \(\chi_{6001}(860,\cdot)\) ...

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

\((2825,3180)\) → \((e\left(\frac{1}{16}\right),e\left(\frac{1}{352}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(1\)\(1\)\(e\left(\frac{15}{44}\right)\)\(e\left(\frac{23}{352}\right)\)\(e\left(\frac{15}{22}\right)\)\(e\left(\frac{67}{352}\right)\)\(e\left(\frac{13}{32}\right)\)\(e\left(\frac{31}{32}\right)\)\(e\left(\frac{1}{44}\right)\)\(e\left(\frac{23}{176}\right)\)\(e\left(\frac{17}{32}\right)\)\(e\left(\frac{107}{176}\right)\)
value at e.g. 2

Related number fields

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