Properties

Label 6017.2375
Modulus $6017$
Conductor $6017$
Order $42$
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(42))
 
M = H._module
 
chi = DirichletCharacter(H, M([21,17]))
 
pari: [g,chi] = znchar(Mod(2375,6017))
 

Basic properties

Modulus: \(6017\)
Conductor: \(6017\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(42\)
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.bd

\(\chi_{6017}(120,\cdot)\) \(\chi_{6017}(351,\cdot)\) \(\chi_{6017}(1627,\cdot)\) \(\chi_{6017}(2375,\cdot)\) \(\chi_{6017}(3156,\cdot)\) \(\chi_{6017}(3321,\cdot)\) \(\chi_{6017}(3651,\cdot)\) \(\chi_{6017}(3706,\cdot)\) \(\chi_{6017}(3816,\cdot)\) \(\chi_{6017}(3871,\cdot)\) \(\chi_{6017}(4806,\cdot)\) \(\chi_{6017}(5301,\cdot)\)

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: \(\Q(\zeta_{21})\)
Fixed field: Number field defined by a degree 42 polynomial

Values on generators

\((3830,2190)\) → \((-1,e\left(\frac{17}{42}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 6017 }(2375, a) \) \(1\)\(1\)\(e\left(\frac{19}{21}\right)\)\(e\left(\frac{9}{14}\right)\)\(e\left(\frac{17}{21}\right)\)\(e\left(\frac{19}{42}\right)\)\(e\left(\frac{23}{42}\right)\)\(e\left(\frac{4}{21}\right)\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{5}{14}\right)\)\(e\left(\frac{19}{42}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6017 }(2375,a) \;\) at \(\;a = \) e.g. 2