Properties

Label 6042.55
Modulus $6042$
Conductor $1007$
Order $468$
Real no
Primitive no
Minimal yes
Parity odd

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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([0,260,9]))
 
pari: [g,chi] = znchar(Mod(55,6042))
 

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 \(\chi_{1007}(55,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6042.cr

\(\chi_{6042}(55,\cdot)\) \(\chi_{6042}(61,\cdot)\) \(\chi_{6042}(73,\cdot)\) \(\chi_{6042}(85,\cdot)\) \(\chi_{6042}(139,\cdot)\) \(\chi_{6042}(157,\cdot)\) \(\chi_{6042}(253,\cdot)\) \(\chi_{6042}(283,\cdot)\) \(\chi_{6042}(313,\cdot)\) \(\chi_{6042}(385,\cdot)\) \(\chi_{6042}(397,\cdot)\) \(\chi_{6042}(403,\cdot)\) \(\chi_{6042}(427,\cdot)\) \(\chi_{6042}(499,\cdot)\) \(\chi_{6042}(511,\cdot)\) \(\chi_{6042}(595,\cdot)\) \(\chi_{6042}(631,\cdot)\) \(\chi_{6042}(655,\cdot)\) \(\chi_{6042}(709,\cdot)\) \(\chi_{6042}(739,\cdot)\) \(\chi_{6042}(745,\cdot)\) \(\chi_{6042}(769,\cdot)\) \(\chi_{6042}(853,\cdot)\) \(\chi_{6042}(883,\cdot)\) \(\chi_{6042}(973,\cdot)\) \(\chi_{6042}(985,\cdot)\) \(\chi_{6042}(1081,\cdot)\) \(\chi_{6042}(1087,\cdot)\) \(\chi_{6042}(1099,\cdot)\) \(\chi_{6042}(1111,\cdot)\) ...

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

\((2015,4771,2281)\) → \((1,e\left(\frac{5}{9}\right),e\left(\frac{1}{52}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(23\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 6042 }(55, a) \) \(-1\)\(1\)\(e\left(\frac{371}{468}\right)\)\(e\left(\frac{47}{78}\right)\)\(e\left(\frac{61}{78}\right)\)\(e\left(\frac{28}{117}\right)\)\(e\left(\frac{175}{234}\right)\)\(e\left(\frac{31}{36}\right)\)\(e\left(\frac{137}{234}\right)\)\(e\left(\frac{77}{234}\right)\)\(e\left(\frac{151}{156}\right)\)\(e\left(\frac{185}{468}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6042 }(55,a) \;\) at \(\;a = \) e.g. 2