Properties

Label 2808.139
Modulus $2808$
Conductor $2808$
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(2808, base_ring=CyclotomicField(18))
 
M = H._module
 
chi = DirichletCharacter(H, M([9,9,2,12]))
 
pari: [g,chi] = znchar(Mod(139,2808))
 

Basic properties

Modulus: \(2808\)
Conductor: \(2808\)
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 2808.gf

\(\chi_{2808}(139,\cdot)\) \(\chi_{2808}(835,\cdot)\) \(\chi_{2808}(1075,\cdot)\) \(\chi_{2808}(1771,\cdot)\) \(\chi_{2808}(2011,\cdot)\) \(\chi_{2808}(2707,\cdot)\)

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

\((703,1405,2081,1081)\) → \((-1,-1,e\left(\frac{1}{9}\right),e\left(\frac{2}{3}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 2808 }(139, a) \) \(-1\)\(1\)\(e\left(\frac{1}{18}\right)\)\(e\left(\frac{11}{18}\right)\)\(e\left(\frac{1}{9}\right)\)\(1\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{7}{18}\right)\)\(e\left(\frac{1}{9}\right)\)\(e\left(\frac{5}{18}\right)\)\(e\left(\frac{13}{18}\right)\)\(e\left(\frac{2}{3}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2808 }(139,a) \;\) at \(\;a = \) e.g. 2