Properties

Label 6017.585
Modulus $6017$
Conductor $6017$
Order $130$
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(130))
 
M = H._module
 
chi = DirichletCharacter(H, M([13,55]))
 
pari: [g,chi] = znchar(Mod(585,6017))
 

Basic properties

Modulus: \(6017\)
Conductor: \(6017\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(130\)
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.bp

\(\chi_{6017}(28,\cdot)\) \(\chi_{6017}(30,\cdot)\) \(\chi_{6017}(72,\cdot)\) \(\chi_{6017}(107,\cdot)\) \(\chi_{6017}(172,\cdot)\) \(\chi_{6017}(194,\cdot)\) \(\chi_{6017}(310,\cdot)\) \(\chi_{6017}(501,\cdot)\) \(\chi_{6017}(585,\cdot)\) \(\chi_{6017}(744,\cdot)\) \(\chi_{6017}(833,\cdot)\) \(\chi_{6017}(1124,\cdot)\) \(\chi_{6017}(1201,\cdot)\) \(\chi_{6017}(1348,\cdot)\) \(\chi_{6017}(1404,\cdot)\) \(\chi_{6017}(1669,\cdot)\) \(\chi_{6017}(1679,\cdot)\) \(\chi_{6017}(1713,\cdot)\) \(\chi_{6017}(1927,\cdot)\) \(\chi_{6017}(2142,\cdot)\) \(\chi_{6017}(2218,\cdot)\) \(\chi_{6017}(2295,\cdot)\) \(\chi_{6017}(2360,\cdot)\) \(\chi_{6017}(2382,\cdot)\) \(\chi_{6017}(2763,\cdot)\) \(\chi_{6017}(2807,\cdot)\) \(\chi_{6017}(2932,\cdot)\) \(\chi_{6017}(2989,\cdot)\) \(\chi_{6017}(3021,\cdot)\) \(\chi_{6017}(3236,\cdot)\) ...

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: $\Q(\zeta_{65})$
Fixed field: Number field defined by a degree 130 polynomial (not computed)

Values on generators

\((3830,2190)\) → \((e\left(\frac{1}{10}\right),e\left(\frac{11}{26}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 6017 }(585, a) \) \(1\)\(1\)\(e\left(\frac{34}{65}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{3}{65}\right)\)\(e\left(\frac{17}{130}\right)\)\(e\left(\frac{107}{130}\right)\)\(e\left(\frac{18}{65}\right)\)\(e\left(\frac{37}{65}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{17}{26}\right)\)\(e\left(\frac{9}{26}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6017 }(585,a) \;\) at \(\;a = \) e.g. 2