Properties

Label 6017.2174
Modulus $6017$
Conductor $6017$
Order $210$
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([147,145]))
 
pari: [g,chi] = znchar(Mod(2174,6017))
 

Basic properties

Modulus: \(6017\)
Conductor: \(6017\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(210\)
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.bu

\(\chi_{6017}(39,\cdot)\) \(\chi_{6017}(369,\cdot)\) \(\chi_{6017}(424,\cdot)\) \(\chi_{6017}(534,\cdot)\) \(\chi_{6017}(589,\cdot)\) \(\chi_{6017}(667,\cdot)\) \(\chi_{6017}(734,\cdot)\) \(\chi_{6017}(898,\cdot)\) \(\chi_{6017}(1080,\cdot)\) \(\chi_{6017}(1515,\cdot)\) \(\chi_{6017}(1524,\cdot)\) \(\chi_{6017}(1680,\cdot)\) \(\chi_{6017}(1828,\cdot)\) \(\chi_{6017}(2010,\cdot)\) \(\chi_{6017}(2019,\cdot)\) \(\chi_{6017}(2065,\cdot)\) \(\chi_{6017}(2174,\cdot)\) \(\chi_{6017}(2175,\cdot)\) \(\chi_{6017}(2230,\cdot)\) \(\chi_{6017}(2609,\cdot)\) \(\chi_{6017}(2774,\cdot)\) \(\chi_{6017}(2855,\cdot)\) \(\chi_{6017}(2922,\cdot)\) \(\chi_{6017}(3086,\cdot)\) \(\chi_{6017}(3104,\cdot)\) \(\chi_{6017}(3159,\cdot)\) \(\chi_{6017}(3165,\cdot)\) \(\chi_{6017}(3269,\cdot)\) \(\chi_{6017}(3324,\cdot)\) \(\chi_{6017}(3660,\cdot)\) ...

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 210 polynomial (not computed)

Values on generators

\((3830,2190)\) → \((e\left(\frac{7}{10}\right),e\left(\frac{29}{42}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 6017 }(2174, a) \) \(1\)\(1\)\(e\left(\frac{41}{105}\right)\)\(e\left(\frac{57}{70}\right)\)\(e\left(\frac{82}{105}\right)\)\(e\left(\frac{83}{210}\right)\)\(e\left(\frac{43}{210}\right)\)\(e\left(\frac{2}{105}\right)\)\(e\left(\frac{6}{35}\right)\)\(e\left(\frac{22}{35}\right)\)\(e\left(\frac{11}{14}\right)\)\(e\left(\frac{25}{42}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6017 }(2174,a) \;\) at \(\;a = \) e.g. 2