Properties

Label 6012.181
Modulus $6012$
Conductor $167$
Order $83$
Real no
Primitive no
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(6012)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,0,79]))
 
pari: [g,chi] = znchar(Mod(181,6012))
 

Basic properties

Modulus: \(6012\)
Conductor: \(167\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(83\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{167}(14,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6012.q

\(\chi_{6012}(181,\cdot)\) \(\chi_{6012}(217,\cdot)\) \(\chi_{6012}(289,\cdot)\) \(\chi_{6012}(361,\cdot)\) \(\chi_{6012}(397,\cdot)\) \(\chi_{6012}(433,\cdot)\) \(\chi_{6012}(505,\cdot)\) \(\chi_{6012}(577,\cdot)\) \(\chi_{6012}(613,\cdot)\) \(\chi_{6012}(757,\cdot)\) \(\chi_{6012}(901,\cdot)\) \(\chi_{6012}(1009,\cdot)\) \(\chi_{6012}(1117,\cdot)\) \(\chi_{6012}(1225,\cdot)\) \(\chi_{6012}(1297,\cdot)\) \(\chi_{6012}(1369,\cdot)\) \(\chi_{6012}(1477,\cdot)\) \(\chi_{6012}(1657,\cdot)\) \(\chi_{6012}(1873,\cdot)\) \(\chi_{6012}(1909,\cdot)\) \(\chi_{6012}(1945,\cdot)\) \(\chi_{6012}(1981,\cdot)\) \(\chi_{6012}(2053,\cdot)\) \(\chi_{6012}(2089,\cdot)\) \(\chi_{6012}(2125,\cdot)\) \(\chi_{6012}(2161,\cdot)\) \(\chi_{6012}(2233,\cdot)\) \(\chi_{6012}(2269,\cdot)\) \(\chi_{6012}(2341,\cdot)\) \(\chi_{6012}(2413,\cdot)\) ...

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

\((3007,3341,4681)\) → \((1,1,e\left(\frac{79}{83}\right))\)

Values

\(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\(1\)\(1\)\(e\left(\frac{79}{83}\right)\)\(e\left(\frac{26}{83}\right)\)\(e\left(\frac{54}{83}\right)\)\(e\left(\frac{3}{83}\right)\)\(e\left(\frac{37}{83}\right)\)\(e\left(\frac{17}{83}\right)\)\(e\left(\frac{19}{83}\right)\)\(e\left(\frac{75}{83}\right)\)\(e\left(\frac{64}{83}\right)\)\(e\left(\frac{55}{83}\right)\)
value at e.g. 2

Related number fields

Field of values: $\Q(\zeta_{83})$
Fixed field: Number field defined by a degree 83 polynomial