Properties

Label 6017.426
Modulus $6017$
Conductor $6017$
Order $390$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6017, base_ring=CyclotomicField(390))
 
M = H._module
 
chi = DirichletCharacter(H, M([117,295]))
 
pari: [g,chi] = znchar(Mod(426,6017))
 

Basic properties

Modulus: \(6017\)
Conductor: \(6017\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(390\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6017.by

\(\chi_{6017}(83,\cdot)\) \(\chi_{6017}(106,\cdot)\) \(\chi_{6017}(128,\cdot)\) \(\chi_{6017}(145,\cdot)\) \(\chi_{6017}(222,\cdot)\) \(\chi_{6017}(314,\cdot)\) \(\chi_{6017}(316,\cdot)\) \(\chi_{6017}(348,\cdot)\) \(\chi_{6017}(426,\cdot)\) \(\chi_{6017}(536,\cdot)\) \(\chi_{6017}(798,\cdot)\) \(\chi_{6017}(855,\cdot)\) \(\chi_{6017}(877,\cdot)\) \(\chi_{6017}(965,\cdot)\) \(\chi_{6017}(998,\cdot)\) \(\chi_{6017}(1040,\cdot)\) \(\chi_{6017}(1047,\cdot)\) \(\chi_{6017}(1073,\cdot)\) \(\chi_{6017}(1196,\cdot)\) \(\chi_{6017}(1239,\cdot)\) \(\chi_{6017}(1316,\cdot)\) \(\chi_{6017}(1339,\cdot)\) \(\chi_{6017}(1410,\cdot)\) \(\chi_{6017}(1460,\cdot)\) \(\chi_{6017}(1520,\cdot)\) \(\chi_{6017}(1630,\cdot)\) \(\chi_{6017}(1667,\cdot)\) \(\chi_{6017}(1700,\cdot)\) \(\chi_{6017}(1724,\cdot)\) \(\chi_{6017}(1949,\cdot)\) ...

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: $\Q(\zeta_{195})$
Fixed field: Number field defined by a degree 390 polynomial (not computed)

Values on generators

\((3830,2190)\) → \((e\left(\frac{3}{10}\right),e\left(\frac{59}{78}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 6017 }(426, a) \) \(1\)\(1\)\(e\left(\frac{11}{195}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{22}{195}\right)\)\(e\left(\frac{233}{390}\right)\)\(e\left(\frac{373}{390}\right)\)\(e\left(\frac{2}{195}\right)\)\(e\left(\frac{11}{65}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{17}{26}\right)\)\(e\left(\frac{1}{78}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6017 }(426,a) \;\) at \(\;a = \) e.g. 2