Properties

Label 6001.9
Modulus $6001$
Conductor $6001$
Order $176$
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(9,6001))
 

Basic properties

Modulus: \(6001\)
Conductor: \(6001\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(176\)
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.ec

\(\chi_{6001}(9,\cdot)\) \(\chi_{6001}(15,\cdot)\) \(\chi_{6001}(25,\cdot)\) \(\chi_{6001}(127,\cdot)\) \(\chi_{6001}(172,\cdot)\) \(\chi_{6001}(338,\cdot)\) \(\chi_{6001}(372,\cdot)\) \(\chi_{6001}(399,\cdot)\) \(\chi_{6001}(542,\cdot)\) \(\chi_{6001}(593,\cdot)\) \(\chi_{6001}(614,\cdot)\) \(\chi_{6001}(620,\cdot)\) \(\chi_{6001}(665,\cdot)\) \(\chi_{6001}(729,\cdot)\) \(\chi_{6001}(1137,\cdot)\) \(\chi_{6001}(1158,\cdot)\) \(\chi_{6001}(1215,\cdot)\) \(\chi_{6001}(1362,\cdot)\) \(\chi_{6001}(1403,\cdot)\) \(\chi_{6001}(1477,\cdot)\) \(\chi_{6001}(1556,\cdot)\) \(\chi_{6001}(1726,\cdot)\) \(\chi_{6001}(1783,\cdot)\) \(\chi_{6001}(1895,\cdot)\) \(\chi_{6001}(1930,\cdot)\) \(\chi_{6001}(1998,\cdot)\) \(\chi_{6001}(2025,\cdot)\) \(\chi_{6001}(2042,\cdot)\) \(\chi_{6001}(2212,\cdot)\) \(\chi_{6001}(2270,\cdot)\) ...

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

\((2825,3180)\) → \((e\left(\frac{1}{8}\right),e\left(\frac{1}{176}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(1\)\(1\)\(e\left(\frac{15}{22}\right)\)\(e\left(\frac{23}{176}\right)\)\(e\left(\frac{4}{11}\right)\)\(e\left(\frac{67}{176}\right)\)\(e\left(\frac{13}{16}\right)\)\(e\left(\frac{15}{16}\right)\)\(e\left(\frac{1}{22}\right)\)\(e\left(\frac{23}{88}\right)\)\(e\left(\frac{1}{16}\right)\)\(e\left(\frac{19}{88}\right)\)
value at e.g. 2

Related number fields

Field of values: $\Q(\zeta_{176})$
Fixed field: Number field defined by a degree 176 polynomial