Properties

Label 6042.473
Modulus $6042$
Conductor $3021$
Order $234$
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(234))
 
M = H._module
 
chi = DirichletCharacter(H, M([117,130,126]))
 
pari: [g,chi] = znchar(Mod(473,6042))
 

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 \(\chi_{3021}(473,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6042.cl

\(\chi_{6042}(47,\cdot)\) \(\chi_{6042}(119,\cdot)\) \(\chi_{6042}(275,\cdot)\) \(\chi_{6042}(365,\cdot)\) \(\chi_{6042}(473,\cdot)\) \(\chi_{6042}(593,\cdot)\) \(\chi_{6042}(731,\cdot)\) \(\chi_{6042}(917,\cdot)\) \(\chi_{6042}(929,\cdot)\) \(\chi_{6042}(947,\cdot)\) \(\chi_{6042}(1031,\cdot)\) \(\chi_{6042}(1043,\cdot)\) \(\chi_{6042}(1049,\cdot)\) \(\chi_{6042}(1073,\cdot)\) \(\chi_{6042}(1157,\cdot)\) \(\chi_{6042}(1391,\cdot)\) \(\chi_{6042}(1499,\cdot)\) \(\chi_{6042}(1583,\cdot)\) \(\chi_{6042}(1745,\cdot)\) \(\chi_{6042}(1871,\cdot)\) \(\chi_{6042}(1955,\cdot)\) \(\chi_{6042}(1985,\cdot)\) \(\chi_{6042}(2183,\cdot)\) \(\chi_{6042}(2189,\cdot)\) \(\chi_{6042}(2201,\cdot)\) \(\chi_{6042}(2303,\cdot)\) \(\chi_{6042}(2315,\cdot)\) \(\chi_{6042}(2381,\cdot)\) \(\chi_{6042}(2429,\cdot)\) \(\chi_{6042}(2639,\cdot)\) ...

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

\((2015,4771,2281)\) → \((-1,e\left(\frac{5}{9}\right),e\left(\frac{7}{13}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(23\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 6042 }(473, a) \) \(-1\)\(1\)\(e\left(\frac{163}{234}\right)\)\(e\left(\frac{34}{39}\right)\)\(e\left(\frac{31}{78}\right)\)\(e\left(\frac{82}{117}\right)\)\(e\left(\frac{103}{234}\right)\)\(e\left(\frac{11}{18}\right)\)\(e\left(\frac{46}{117}\right)\)\(e\left(\frac{167}{234}\right)\)\(e\left(\frac{4}{39}\right)\)\(e\left(\frac{133}{234}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6042 }(473,a) \;\) at \(\;a = \) e.g. 2