Properties

Label 6042.4181
Modulus $6042$
Conductor $159$
Order $26$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6042, base_ring=CyclotomicField(26))
 
M = H._module
 
chi = DirichletCharacter(H, M([13,0,22]))
 
pari: [g,chi] = znchar(Mod(4181,6042))
 

Basic properties

Modulus: \(6042\)
Conductor: \(159\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(26\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{159}(47,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6042.bm

\(\chi_{6042}(77,\cdot)\) \(\chi_{6042}(1901,\cdot)\) \(\chi_{6042}(2699,\cdot)\) \(\chi_{6042}(3155,\cdot)\) \(\chi_{6042}(3269,\cdot)\) \(\chi_{6042}(3383,\cdot)\) \(\chi_{6042}(3725,\cdot)\) \(\chi_{6042}(4181,\cdot)\) \(\chi_{6042}(4409,\cdot)\) \(\chi_{6042}(4865,\cdot)\) \(\chi_{6042}(5207,\cdot)\) \(\chi_{6042}(6005,\cdot)\)

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: \(\Q(\zeta_{13})\)
Fixed field: 26.0.384766437057818380952237905666104641217782272563.1

Values on generators

\((2015,4771,2281)\) → \((-1,1,e\left(\frac{11}{13}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(23\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 6042 }(4181, a) \) \(-1\)\(1\)\(e\left(\frac{7}{26}\right)\)\(e\left(\frac{11}{13}\right)\)\(e\left(\frac{15}{26}\right)\)\(e\left(\frac{4}{13}\right)\)\(e\left(\frac{25}{26}\right)\)\(-1\)\(e\left(\frac{7}{13}\right)\)\(e\left(\frac{11}{26}\right)\)\(e\left(\frac{12}{13}\right)\)\(e\left(\frac{3}{26}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6042 }(4181,a) \;\) at \(\;a = \) e.g. 2