Properties

Label 6001.37
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([22,107]))
 
pari: [g,chi] = znchar(Mod(37,6001))
 

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.ef

\(\chi_{6001}(37,\cdot)\) \(\chi_{6001}(126,\cdot)\) \(\chi_{6001}(158,\cdot)\) \(\chi_{6001}(160,\cdot)\) \(\chi_{6001}(249,\cdot)\) \(\chi_{6001}(258,\cdot)\) \(\chi_{6001}(266,\cdot)\) \(\chi_{6001}(350,\cdot)\) \(\chi_{6001}(401,\cdot)\) \(\chi_{6001}(428,\cdot)\) \(\chi_{6001}(456,\cdot)\) \(\chi_{6001}(471,\cdot)\) \(\chi_{6001}(507,\cdot)\) \(\chi_{6001}(556,\cdot)\) \(\chi_{6001}(558,\cdot)\) \(\chi_{6001}(598,\cdot)\) \(\chi_{6001}(734,\cdot)\) \(\chi_{6001}(777,\cdot)\) \(\chi_{6001}(793,\cdot)\) \(\chi_{6001}(810,\cdot)\) \(\chi_{6001}(813,\cdot)\) \(\chi_{6001}(879,\cdot)\) \(\chi_{6001}(896,\cdot)\) \(\chi_{6001}(921,\cdot)\) \(\chi_{6001}(940,\cdot)\) \(\chi_{6001}(1178,\cdot)\) \(\chi_{6001}(1269,\cdot)\) \(\chi_{6001}(1287,\cdot)\) \(\chi_{6001}(1332,\cdot)\) \(\chi_{6001}(1338,\cdot)\) ...

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

\((2825,3180)\) → \((e\left(\frac{1}{16}\right),e\left(\frac{107}{352}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(1\)\(1\)\(e\left(\frac{8}{11}\right)\)\(e\left(\frac{129}{352}\right)\)\(e\left(\frac{5}{11}\right)\)\(e\left(\frac{85}{352}\right)\)\(e\left(\frac{3}{32}\right)\)\(e\left(\frac{25}{32}\right)\)\(e\left(\frac{2}{11}\right)\)\(e\left(\frac{129}{176}\right)\)\(e\left(\frac{31}{32}\right)\)\(e\left(\frac{119}{176}\right)\)
value at e.g. 2

Related number fields

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