Properties

Label 6042.191
Modulus $6042$
Conductor $159$
Order $52$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6042, base_ring=CyclotomicField(52))
 
M = H._module
 
chi = DirichletCharacter(H, M([26,0,5]))
 
pari: [g,chi] = znchar(Mod(191,6042))
 

Basic properties

Modulus: \(6042\)
Conductor: \(159\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(52\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{159}(32,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6042.bt

\(\chi_{6042}(191,\cdot)\) \(\chi_{6042}(419,\cdot)\) \(\chi_{6042}(533,\cdot)\) \(\chi_{6042}(761,\cdot)\) \(\chi_{6042}(875,\cdot)\) \(\chi_{6042}(989,\cdot)\) \(\chi_{6042}(1217,\cdot)\) \(\chi_{6042}(1445,\cdot)\) \(\chi_{6042}(1559,\cdot)\) \(\chi_{6042}(2471,\cdot)\) \(\chi_{6042}(2585,\cdot)\) \(\chi_{6042}(2927,\cdot)\) \(\chi_{6042}(3041,\cdot)\) \(\chi_{6042}(3953,\cdot)\) \(\chi_{6042}(4067,\cdot)\) \(\chi_{6042}(4295,\cdot)\) \(\chi_{6042}(4523,\cdot)\) \(\chi_{6042}(4637,\cdot)\) \(\chi_{6042}(4751,\cdot)\) \(\chi_{6042}(4979,\cdot)\) \(\chi_{6042}(5093,\cdot)\) \(\chi_{6042}(5321,\cdot)\) \(\chi_{6042}(5663,\cdot)\) \(\chi_{6042}(5891,\cdot)\)

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: $\Q(\zeta_{52})$
Fixed field: Number field defined by a degree 52 polynomial

Values on generators

\((2015,4771,2281)\) → \((-1,1,e\left(\frac{5}{52}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(23\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 6042 }(191, a) \) \(1\)\(1\)\(e\left(\frac{1}{52}\right)\)\(e\left(\frac{9}{26}\right)\)\(e\left(\frac{1}{13}\right)\)\(e\left(\frac{4}{13}\right)\)\(e\left(\frac{6}{13}\right)\)\(i\)\(e\left(\frac{1}{26}\right)\)\(e\left(\frac{12}{13}\right)\)\(e\left(\frac{9}{52}\right)\)\(e\left(\frac{19}{52}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6042 }(191,a) \;\) at \(\;a = \) e.g. 2