Properties

Label 6012.1889
Modulus $6012$
Conductor $501$
Order $166$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6012, base_ring=CyclotomicField(166))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,83,17]))
 
pari: [g,chi] = znchar(Mod(1889,6012))
 

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 \(\chi_{501}(386,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
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()
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: $\Q(\zeta_{83})$
Fixed field: Number field defined by a degree 166 polynomial (not computed)

Values on generators

\((3007,3341,4681)\) → \((1,-1,e\left(\frac{17}{166}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 6012 }(1889, a) \) \(1\)\(1\)\(e\left(\frac{50}{83}\right)\)\(e\left(\frac{7}{83}\right)\)\(e\left(\frac{61}{166}\right)\)\(e\left(\frac{91}{166}\right)\)\(e\left(\frac{77}{83}\right)\)\(e\left(\frac{78}{83}\right)\)\(e\left(\frac{53}{83}\right)\)\(e\left(\frac{17}{83}\right)\)\(e\left(\frac{143}{166}\right)\)\(e\left(\frac{18}{83}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6012 }(1889,a) \;\) at \(\;a = \) e.g. 2