Properties

Label 6012.1351
Modulus $6012$
Conductor $668$
Order $166$
Real no
Primitive no
Minimal yes
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([83,0,95]))
 
pari: [g,chi] = znchar(Mod(1351,6012))
 

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 \(\chi_{668}(15,\cdot)\)
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()
 
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

\((3007,3341,4681)\) → \((-1,1,e\left(\frac{95}{166}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 6012 }(1351, a) \) \(1\)\(1\)\(e\left(\frac{95}{166}\right)\)\(e\left(\frac{5}{166}\right)\)\(e\left(\frac{87}{166}\right)\)\(e\left(\frac{157}{166}\right)\)\(e\left(\frac{55}{166}\right)\)\(e\left(\frac{115}{166}\right)\)\(e\left(\frac{13}{83}\right)\)\(e\left(\frac{12}{83}\right)\)\(e\left(\frac{70}{83}\right)\)\(e\left(\frac{1}{166}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6012 }(1351,a) \;\) at \(\;a = \) e.g. 2