Properties

Label 6069.188
Modulus $6069$
Conductor $6069$
Order $34$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6069, base_ring=CyclotomicField(34))
 
M = H._module
 
chi = DirichletCharacter(H, M([17,17,26]))
 
pari: [g,chi] = znchar(Mod(188,6069))
 

Basic properties

Modulus: \(6069\)
Conductor: \(6069\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(34\)
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 6069.bp

\(\chi_{6069}(188,\cdot)\) \(\chi_{6069}(545,\cdot)\) \(\chi_{6069}(902,\cdot)\) \(\chi_{6069}(1259,\cdot)\) \(\chi_{6069}(1616,\cdot)\) \(\chi_{6069}(1973,\cdot)\) \(\chi_{6069}(2330,\cdot)\) \(\chi_{6069}(2687,\cdot)\) \(\chi_{6069}(3044,\cdot)\) \(\chi_{6069}(3401,\cdot)\) \(\chi_{6069}(4115,\cdot)\) \(\chi_{6069}(4472,\cdot)\) \(\chi_{6069}(4829,\cdot)\) \(\chi_{6069}(5186,\cdot)\) \(\chi_{6069}(5543,\cdot)\) \(\chi_{6069}(5900,\cdot)\)

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: \(\Q(\zeta_{17})\)
Fixed field: Number field defined by a degree 34 polynomial

Values on generators

\((2024,4336,3760)\) → \((-1,-1,e\left(\frac{13}{17}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(19\)\(20\)
\( \chi_{ 6069 }(188, a) \) \(1\)\(1\)\(e\left(\frac{27}{34}\right)\)\(e\left(\frac{10}{17}\right)\)\(e\left(\frac{2}{17}\right)\)\(e\left(\frac{13}{34}\right)\)\(e\left(\frac{31}{34}\right)\)\(e\left(\frac{3}{34}\right)\)\(e\left(\frac{13}{34}\right)\)\(e\left(\frac{3}{17}\right)\)\(e\left(\frac{7}{34}\right)\)\(e\left(\frac{12}{17}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6069 }(188,a) \;\) at \(\;a = \) e.g. 2