Properties

Label 6042.5
Modulus $6042$
Conductor $3021$
Order $468$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6042, base_ring=CyclotomicField(468))
 
M = H._module
 
chi = DirichletCharacter(H, M([234,416,423]))
 
pari: [g,chi] = znchar(Mod(5,6042))
 

Basic properties

Modulus: \(6042\)
Conductor: \(3021\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(468\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{3021}(5,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6042.ct

\(\chi_{6042}(5,\cdot)\) \(\chi_{6042}(35,\cdot)\) \(\chi_{6042}(101,\cdot)\) \(\chi_{6042}(137,\cdot)\) \(\chi_{6042}(161,\cdot)\) \(\chi_{6042}(215,\cdot)\) \(\chi_{6042}(233,\cdot)\) \(\chi_{6042}(245,\cdot)\) \(\chi_{6042}(251,\cdot)\) \(\chi_{6042}(263,\cdot)\) \(\chi_{6042}(359,\cdot)\) \(\chi_{6042}(389,\cdot)\) \(\chi_{6042}(443,\cdot)\) \(\chi_{6042}(479,\cdot)\) \(\chi_{6042}(491,\cdot)\) \(\chi_{6042}(503,\cdot)\) \(\chi_{6042}(557,\cdot)\) \(\chi_{6042}(575,\cdot)\) \(\chi_{6042}(605,\cdot)\) \(\chi_{6042}(617,\cdot)\) \(\chi_{6042}(671,\cdot)\) \(\chi_{6042}(701,\cdot)\) \(\chi_{6042}(707,\cdot)\) \(\chi_{6042}(803,\cdot)\) \(\chi_{6042}(815,\cdot)\) \(\chi_{6042}(821,\cdot)\) \(\chi_{6042}(845,\cdot)\) \(\chi_{6042}(899,\cdot)\) \(\chi_{6042}(935,\cdot)\) \(\chi_{6042}(959,\cdot)\) ...

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: $\Q(\zeta_{468})$
Fixed field: Number field defined by a degree 468 polynomial (not computed)

Values on generators

\((2015,4771,2281)\) → \((-1,e\left(\frac{8}{9}\right),e\left(\frac{47}{52}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(23\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 6042 }(5, a) \) \(1\)\(1\)\(e\left(\frac{95}{468}\right)\)\(e\left(\frac{77}{78}\right)\)\(e\left(\frac{23}{39}\right)\)\(e\left(\frac{16}{117}\right)\)\(e\left(\frac{50}{117}\right)\)\(e\left(\frac{19}{36}\right)\)\(e\left(\frac{95}{234}\right)\)\(e\left(\frac{22}{117}\right)\)\(e\left(\frac{25}{156}\right)\)\(e\left(\frac{89}{468}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6042 }(5,a) \;\) at \(\;a = \) e.g. 2