Properties

Label 6042.17
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([117,130,45]))
 
pari: [g,chi] = znchar(Mod(17,6042))
 

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 \(\chi_{3021}(17,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6042.cj

\(\chi_{6042}(17,\cdot)\) \(\chi_{6042}(131,\cdot)\) \(\chi_{6042}(149,\cdot)\) \(\chi_{6042}(329,\cdot)\) \(\chi_{6042}(347,\cdot)\) \(\chi_{6042}(377,\cdot)\) \(\chi_{6042}(461,\cdot)\) \(\chi_{6042}(587,\cdot)\) \(\chi_{6042}(785,\cdot)\) \(\chi_{6042}(833,\cdot)\) \(\chi_{6042}(1013,\cdot)\) \(\chi_{6042}(1175,\cdot)\) \(\chi_{6042}(1259,\cdot)\) \(\chi_{6042}(1289,\cdot)\) \(\chi_{6042}(1301,\cdot)\) \(\chi_{6042}(1385,\cdot)\) \(\chi_{6042}(1403,\cdot)\) \(\chi_{6042}(1415,\cdot)\) \(\chi_{6042}(1469,\cdot)\) \(\chi_{6042}(1601,\cdot)\) \(\chi_{6042}(1619,\cdot)\) \(\chi_{6042}(1733,\cdot)\) \(\chi_{6042}(1811,\cdot)\) \(\chi_{6042}(1859,\cdot)\) \(\chi_{6042}(1925,\cdot)\) \(\chi_{6042}(2039,\cdot)\) \(\chi_{6042}(2057,\cdot)\) \(\chi_{6042}(2213,\cdot)\) \(\chi_{6042}(2285,\cdot)\) \(\chi_{6042}(2495,\cdot)\) ...

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

\((2015,4771,2281)\) → \((-1,e\left(\frac{5}{9}\right),e\left(\frac{5}{26}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(23\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 6042 }(17, a) \) \(-1\)\(1\)\(e\left(\frac{50}{117}\right)\)\(e\left(\frac{1}{39}\right)\)\(e\left(\frac{25}{78}\right)\)\(e\left(\frac{46}{117}\right)\)\(e\left(\frac{229}{234}\right)\)\(e\left(\frac{1}{9}\right)\)\(e\left(\frac{100}{117}\right)\)\(e\left(\frac{185}{234}\right)\)\(e\left(\frac{53}{78}\right)\)\(e\left(\frac{53}{117}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6042 }(17,a) \;\) at \(\;a = \) e.g. 2