Properties

Label 6042.565
Modulus $6042$
Conductor $1007$
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([0,182,81]))
 
pari: [g,chi] = znchar(Mod(565,6042))
 

Basic properties

Modulus: \(6042\)
Conductor: \(1007\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(468\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1007}(565,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6042.cq

\(\chi_{6042}(67,\cdot)\) \(\chi_{6042}(79,\cdot)\) \(\chi_{6042}(109,\cdot)\) \(\chi_{6042}(127,\cdot)\) \(\chi_{6042}(181,\cdot)\) \(\chi_{6042}(193,\cdot)\) \(\chi_{6042}(337,\cdot)\) \(\chi_{6042}(421,\cdot)\) \(\chi_{6042}(451,\cdot)\) \(\chi_{6042}(469,\cdot)\) \(\chi_{6042}(535,\cdot)\) \(\chi_{6042}(565,\cdot)\) \(\chi_{6042}(667,\cdot)\) \(\chi_{6042}(697,\cdot)\) \(\chi_{6042}(763,\cdot)\) \(\chi_{6042}(775,\cdot)\) \(\chi_{6042}(781,\cdot)\) \(\chi_{6042}(793,\cdot)\) \(\chi_{6042}(889,\cdot)\) \(\chi_{6042}(1009,\cdot)\) \(\chi_{6042}(1021,\cdot)\) \(\chi_{6042}(1039,\cdot)\) \(\chi_{6042}(1093,\cdot)\) \(\chi_{6042}(1105,\cdot)\) \(\chi_{6042}(1135,\cdot)\) \(\chi_{6042}(1207,\cdot)\) \(\chi_{6042}(1231,\cdot)\) \(\chi_{6042}(1237,\cdot)\) \(\chi_{6042}(1267,\cdot)\) \(\chi_{6042}(1333,\cdot)\) ...

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

\((2015,4771,2281)\) → \((1,e\left(\frac{7}{18}\right),e\left(\frac{9}{52}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(23\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 6042 }(565, a) \) \(1\)\(1\)\(e\left(\frac{167}{468}\right)\)\(e\left(\frac{59}{78}\right)\)\(e\left(\frac{55}{78}\right)\)\(e\left(\frac{23}{234}\right)\)\(e\left(\frac{145}{234}\right)\)\(e\left(\frac{19}{36}\right)\)\(e\left(\frac{167}{234}\right)\)\(e\left(\frac{67}{117}\right)\)\(e\left(\frac{85}{156}\right)\)\(e\left(\frac{53}{468}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6042 }(565,a) \;\) at \(\;a = \) e.g. 2