Properties

Label 1368.43
Modulus $1368$
Conductor $1368$
Order $18$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1368, base_ring=CyclotomicField(18))
 
M = H._module
 
chi = DirichletCharacter(H, M([9,9,12,16]))
 
pari: [g,chi] = znchar(Mod(43,1368))
 

Basic properties

Modulus: \(1368\)
Conductor: \(1368\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(18\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1368.dh

\(\chi_{1368}(43,\cdot)\) \(\chi_{1368}(139,\cdot)\) \(\chi_{1368}(187,\cdot)\) \(\chi_{1368}(427,\cdot)\) \(\chi_{1368}(859,\cdot)\) \(\chi_{1368}(1195,\cdot)\)

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: \(\Q(\zeta_{9})\)
Fixed field: Number field defined by a degree 18 polynomial

Values on generators

\((343,685,1217,1009)\) → \((-1,-1,e\left(\frac{2}{3}\right),e\left(\frac{8}{9}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(23\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 1368 }(43, a) \) \(-1\)\(1\)\(e\left(\frac{1}{18}\right)\)\(-1\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{5}{18}\right)\)\(e\left(\frac{8}{9}\right)\)\(e\left(\frac{11}{18}\right)\)\(e\left(\frac{1}{9}\right)\)\(e\left(\frac{5}{18}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{5}{9}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1368 }(43,a) \;\) at \(\;a = \) e.g. 2