Properties

Label 6017.2693
Modulus $6017$
Conductor $6017$
Order $105$
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(210))
 
M = H._module
 
chi = DirichletCharacter(H, M([126,100]))
 
pari: [g,chi] = znchar(Mod(2693,6017))
 

Basic properties

Modulus: \(6017\)
Conductor: \(6017\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(105\)
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.bm

\(\chi_{6017}(14,\cdot)\) \(\chi_{6017}(126,\cdot)\) \(\chi_{6017}(169,\cdot)\) \(\chi_{6017}(196,\cdot)\) \(\chi_{6017}(427,\cdot)\) \(\chi_{6017}(664,\cdot)\) \(\chi_{6017}(907,\cdot)\) \(\chi_{6017}(1263,\cdot)\) \(\chi_{6017}(1290,\cdot)\) \(\chi_{6017}(1521,\cdot)\) \(\chi_{6017}(1599,\cdot)\) \(\chi_{6017}(1654,\cdot)\) \(\chi_{6017}(1655,\cdot)\) \(\chi_{6017}(1758,\cdot)\) \(\chi_{6017}(1764,\cdot)\) \(\chi_{6017}(1819,\cdot)\) \(\chi_{6017}(2149,\cdot)\) \(\chi_{6017}(2314,\cdot)\) \(\chi_{6017}(2357,\cdot)\) \(\chi_{6017}(2693,\cdot)\) \(\chi_{6017}(2748,\cdot)\) \(\chi_{6017}(2852,\cdot)\) \(\chi_{6017}(2858,\cdot)\) \(\chi_{6017}(2913,\cdot)\) \(\chi_{6017}(2931,\cdot)\) \(\chi_{6017}(3095,\cdot)\) \(\chi_{6017}(3162,\cdot)\) \(\chi_{6017}(3243,\cdot)\) \(\chi_{6017}(3408,\cdot)\) \(\chi_{6017}(3787,\cdot)\) ...

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: $\Q(\zeta_{105})$
Fixed field: Number field defined by a degree 105 polynomial (not computed)

Values on generators

\((3830,2190)\) → \((e\left(\frac{3}{5}\right),e\left(\frac{10}{21}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 6017 }(2693, a) \) \(1\)\(1\)\(e\left(\frac{8}{105}\right)\)\(e\left(\frac{3}{35}\right)\)\(e\left(\frac{16}{105}\right)\)\(e\left(\frac{67}{105}\right)\)\(e\left(\frac{17}{105}\right)\)\(e\left(\frac{26}{105}\right)\)\(e\left(\frac{8}{35}\right)\)\(e\left(\frac{6}{35}\right)\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{5}{21}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6017 }(2693,a) \;\) at \(\;a = \) e.g. 2