Properties

Label 6042.1133
Modulus $6042$
Conductor $3021$
Order $156$
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(156))
 
M = H._module
 
chi = DirichletCharacter(H, M([78,130,147]))
 
pari: [g,chi] = znchar(Mod(1133,6042))
 

Basic properties

Modulus: \(6042\)
Conductor: \(3021\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(156\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{3021}(1133,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6042.cf

\(\chi_{6042}(65,\cdot)\) \(\chi_{6042}(179,\cdot)\) \(\chi_{6042}(563,\cdot)\) \(\chi_{6042}(677,\cdot)\) \(\chi_{6042}(1019,\cdot)\) \(\chi_{6042}(1091,\cdot)\) \(\chi_{6042}(1133,\cdot)\) \(\chi_{6042}(1205,\cdot)\) \(\chi_{6042}(1433,\cdot)\) \(\chi_{6042}(1661,\cdot)\) \(\chi_{6042}(1775,\cdot)\) \(\chi_{6042}(1889,\cdot)\) \(\chi_{6042}(2045,\cdot)\) \(\chi_{6042}(2117,\cdot)\) \(\chi_{6042}(2159,\cdot)\) \(\chi_{6042}(2231,\cdot)\) \(\chi_{6042}(2387,\cdot)\) \(\chi_{6042}(2459,\cdot)\) \(\chi_{6042}(2615,\cdot)\) \(\chi_{6042}(2729,\cdot)\) \(\chi_{6042}(2801,\cdot)\) \(\chi_{6042}(2843,\cdot)\) \(\chi_{6042}(3029,\cdot)\) \(\chi_{6042}(3071,\cdot)\) \(\chi_{6042}(3185,\cdot)\) \(\chi_{6042}(3371,\cdot)\) \(\chi_{6042}(3413,\cdot)\) \(\chi_{6042}(3599,\cdot)\) \(\chi_{6042}(3713,\cdot)\) \(\chi_{6042}(3755,\cdot)\) ...

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: $\Q(\zeta_{156})$
Fixed field: Number field defined by a degree 156 polynomial (not computed)

Values on generators

\((2015,4771,2281)\) → \((-1,e\left(\frac{5}{6}\right),e\left(\frac{49}{52}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(23\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 6042 }(1133, a) \) \(-1\)\(1\)\(e\left(\frac{19}{156}\right)\)\(e\left(\frac{5}{26}\right)\)\(e\left(\frac{2}{13}\right)\)\(e\left(\frac{61}{78}\right)\)\(e\left(\frac{10}{39}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{19}{78}\right)\)\(e\left(\frac{1}{78}\right)\)\(e\left(\frac{31}{52}\right)\)\(e\left(\frac{49}{156}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6042 }(1133,a) \;\) at \(\;a = \) e.g. 2