Properties

Label 6001.38
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([132,117]))
 
pari: [g,chi] = znchar(Mod(38,6001))
 

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.ea

\(\chi_{6001}(38,\cdot)\) \(\chi_{6001}(200,\cdot)\) \(\chi_{6001}(344,\cdot)\) \(\chi_{6001}(378,\cdot)\) \(\chi_{6001}(480,\cdot)\) \(\chi_{6001}(497,\cdot)\) \(\chi_{6001}(506,\cdot)\) \(\chi_{6001}(667,\cdot)\) \(\chi_{6001}(752,\cdot)\) \(\chi_{6001}(778,\cdot)\) \(\chi_{6001}(863,\cdot)\) \(\chi_{6001}(871,\cdot)\) \(\chi_{6001}(939,\cdot)\) \(\chi_{6001}(965,\cdot)\) \(\chi_{6001}(973,\cdot)\) \(\chi_{6001}(1016,\cdot)\) \(\chi_{6001}(1135,\cdot)\) \(\chi_{6001}(1152,\cdot)\) \(\chi_{6001}(1211,\cdot)\) \(\chi_{6001}(1228,\cdot)\) \(\chi_{6001}(1245,\cdot)\) \(\chi_{6001}(1330,\cdot)\) \(\chi_{6001}(1568,\cdot)\) \(\chi_{6001}(1747,\cdot)\) \(\chi_{6001}(1815,\cdot)\) \(\chi_{6001}(2019,\cdot)\) \(\chi_{6001}(2053,\cdot)\) \(\chi_{6001}(2248,\cdot)\) \(\chi_{6001}(2316,\cdot)\) \(\chi_{6001}(2393,\cdot)\) ...

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

\((2825,3180)\) → \((-i,e\left(\frac{117}{176}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(1\)\(1\)\(e\left(\frac{23}{44}\right)\)\(e\left(\frac{73}{176}\right)\)\(e\left(\frac{1}{22}\right)\)\(e\left(\frac{29}{176}\right)\)\(e\left(\frac{15}{16}\right)\)\(e\left(\frac{1}{16}\right)\)\(e\left(\frac{25}{44}\right)\)\(e\left(\frac{73}{88}\right)\)\(e\left(\frac{11}{16}\right)\)\(e\left(\frac{3}{22}\right)\)
value at e.g. 2

Related number fields

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