Properties

Label 6012.451
Modulus $6012$
Conductor $668$
Order $166$
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([83,0,125]))
 
pari: [g,chi] = znchar(Mod(451,6012))
 

Basic properties

Modulus: \(6012\)
Conductor: \(668\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(166\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{668}(451,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6012.x

\(\chi_{6012}(55,\cdot)\) \(\chi_{6012}(91,\cdot)\) \(\chi_{6012}(163,\cdot)\) \(\chi_{6012}(235,\cdot)\) \(\chi_{6012}(271,\cdot)\) \(\chi_{6012}(307,\cdot)\) \(\chi_{6012}(379,\cdot)\) \(\chi_{6012}(451,\cdot)\) \(\chi_{6012}(487,\cdot)\) \(\chi_{6012}(703,\cdot)\) \(\chi_{6012}(739,\cdot)\) \(\chi_{6012}(811,\cdot)\) \(\chi_{6012}(955,\cdot)\) \(\chi_{6012}(991,\cdot)\) \(\chi_{6012}(1243,\cdot)\) \(\chi_{6012}(1279,\cdot)\) \(\chi_{6012}(1315,\cdot)\) \(\chi_{6012}(1351,\cdot)\) \(\chi_{6012}(1387,\cdot)\) \(\chi_{6012}(1459,\cdot)\) \(\chi_{6012}(1495,\cdot)\) \(\chi_{6012}(1639,\cdot)\) \(\chi_{6012}(1675,\cdot)\) \(\chi_{6012}(1711,\cdot)\) \(\chi_{6012}(1783,\cdot)\) \(\chi_{6012}(1819,\cdot)\) \(\chi_{6012}(1927,\cdot)\) \(\chi_{6012}(2071,\cdot)\) \(\chi_{6012}(2107,\cdot)\) \(\chi_{6012}(2143,\cdot)\) ...

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

\((3007,3341,4681)\) → \((-1,1,e\left(\frac{125}{166}\right))\)

Values

\(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\(1\)\(1\)\(e\left(\frac{125}{166}\right)\)\(e\left(\frac{59}{166}\right)\)\(e\left(\frac{97}{166}\right)\)\(e\left(\frac{93}{166}\right)\)\(e\left(\frac{151}{166}\right)\)\(e\left(\frac{29}{166}\right)\)\(e\left(\frac{4}{83}\right)\)\(e\left(\frac{42}{83}\right)\)\(e\left(\frac{79}{83}\right)\)\(e\left(\frac{45}{166}\right)\)
value at e.g. 2

Related number fields

Field of values: $\Q(\zeta_{83})$
Fixed field: Number field defined by a degree 166 polynomial