Properties

Label 6042.281
Modulus $6042$
Conductor $3021$
Order $234$
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(234))
 
M = H._module
 
chi = DirichletCharacter(H, M([117,143,18]))
 
pari: [g,chi] = znchar(Mod(281,6042))
 

Basic properties

Modulus: \(6042\)
Conductor: \(3021\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(234\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{3021}(281,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6042.ck

\(\chi_{6042}(89,\cdot)\) \(\chi_{6042}(155,\cdot)\) \(\chi_{6042}(203,\cdot)\) \(\chi_{6042}(281,\cdot)\) \(\chi_{6042}(395,\cdot)\) \(\chi_{6042}(413,\cdot)\) \(\chi_{6042}(545,\cdot)\) \(\chi_{6042}(599,\cdot)\) \(\chi_{6042}(611,\cdot)\) \(\chi_{6042}(629,\cdot)\) \(\chi_{6042}(713,\cdot)\) \(\chi_{6042}(725,\cdot)\) \(\chi_{6042}(755,\cdot)\) \(\chi_{6042}(839,\cdot)\) \(\chi_{6042}(1001,\cdot)\) \(\chi_{6042}(1181,\cdot)\) \(\chi_{6042}(1229,\cdot)\) \(\chi_{6042}(1427,\cdot)\) \(\chi_{6042}(1553,\cdot)\) \(\chi_{6042}(1637,\cdot)\) \(\chi_{6042}(1667,\cdot)\) \(\chi_{6042}(1685,\cdot)\) \(\chi_{6042}(1865,\cdot)\) \(\chi_{6042}(1883,\cdot)\) \(\chi_{6042}(1997,\cdot)\) \(\chi_{6042}(2027,\cdot)\) \(\chi_{6042}(2111,\cdot)\) \(\chi_{6042}(2219,\cdot)\) \(\chi_{6042}(2321,\cdot)\) \(\chi_{6042}(2453,\cdot)\) ...

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: $\Q(\zeta_{117})$
Fixed field: Number field defined by a degree 234 polynomial (not computed)

Values on generators

\((2015,4771,2281)\) → \((-1,e\left(\frac{11}{18}\right),e\left(\frac{1}{13}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(23\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 6042 }(281, a) \) \(1\)\(1\)\(e\left(\frac{209}{234}\right)\)\(e\left(\frac{29}{39}\right)\)\(e\left(\frac{23}{78}\right)\)\(e\left(\frac{211}{234}\right)\)\(e\left(\frac{89}{234}\right)\)\(e\left(\frac{13}{18}\right)\)\(e\left(\frac{92}{117}\right)\)\(e\left(\frac{50}{117}\right)\)\(e\left(\frac{55}{78}\right)\)\(e\left(\frac{149}{234}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6042 }(281,a) \;\) at \(\;a = \) e.g. 2