Properties

Label 6012.17
Modulus $6012$
Conductor $501$
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([0,83,53]))
 
pari: [g,chi] = znchar(Mod(17,6012))
 

Basic properties

Modulus: \(6012\)
Conductor: \(501\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(166\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{501}(17,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6012.r

\(\chi_{6012}(17,\cdot)\) \(\chi_{6012}(53,\cdot)\) \(\chi_{6012}(125,\cdot)\) \(\chi_{6012}(161,\cdot)\) \(\chi_{6012}(197,\cdot)\) \(\chi_{6012}(269,\cdot)\) \(\chi_{6012}(305,\cdot)\) \(\chi_{6012}(377,\cdot)\) \(\chi_{6012}(413,\cdot)\) \(\chi_{6012}(485,\cdot)\) \(\chi_{6012}(521,\cdot)\) \(\chi_{6012}(593,\cdot)\) \(\chi_{6012}(665,\cdot)\) \(\chi_{6012}(737,\cdot)\) \(\chi_{6012}(773,\cdot)\) \(\chi_{6012}(845,\cdot)\) \(\chi_{6012}(881,\cdot)\) \(\chi_{6012}(917,\cdot)\) \(\chi_{6012}(953,\cdot)\) \(\chi_{6012}(1025,\cdot)\) \(\chi_{6012}(1061,\cdot)\) \(\chi_{6012}(1097,\cdot)\) \(\chi_{6012}(1133,\cdot)\) \(\chi_{6012}(1349,\cdot)\) \(\chi_{6012}(1529,\cdot)\) \(\chi_{6012}(1637,\cdot)\) \(\chi_{6012}(1709,\cdot)\) \(\chi_{6012}(1781,\cdot)\) \(\chi_{6012}(1889,\cdot)\) \(\chi_{6012}(1997,\cdot)\) ...

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

\((3007,3341,4681)\) → \((1,-1,e\left(\frac{53}{166}\right))\)

Values

\(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\(1\)\(1\)\(e\left(\frac{68}{83}\right)\)\(e\left(\frac{56}{83}\right)\)\(e\left(\frac{73}{166}\right)\)\(e\left(\frac{147}{166}\right)\)\(e\left(\frac{35}{83}\right)\)\(e\left(\frac{43}{83}\right)\)\(e\left(\frac{9}{83}\right)\)\(e\left(\frac{53}{83}\right)\)\(e\left(\frac{65}{166}\right)\)\(e\left(\frac{61}{83}\right)\)
value at e.g. 2

Related number fields

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