Properties

Label 109.9
Modulus $109$
Conductor $109$
Order $27$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(109, base_ring=CyclotomicField(54))
 
M = H._module
 
chi = DirichletCharacter(H, M([52]))
 
pari: [g,chi] = znchar(Mod(9,109))
 

Basic properties

Modulus: \(109\)
Conductor: \(109\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(27\)
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 109.i

\(\chi_{109}(3,\cdot)\) \(\chi_{109}(5,\cdot)\) \(\chi_{109}(7,\cdot)\) \(\chi_{109}(9,\cdot)\) \(\chi_{109}(15,\cdot)\) \(\chi_{109}(21,\cdot)\) \(\chi_{109}(22,\cdot)\) \(\chi_{109}(25,\cdot)\) \(\chi_{109}(26,\cdot)\) \(\chi_{109}(35,\cdot)\) \(\chi_{109}(48,\cdot)\) \(\chi_{109}(49,\cdot)\) \(\chi_{109}(73,\cdot)\) \(\chi_{109}(78,\cdot)\) \(\chi_{109}(80,\cdot)\) \(\chi_{109}(81,\cdot)\) \(\chi_{109}(89,\cdot)\) \(\chi_{109}(97,\cdot)\)

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: \(\Q(\zeta_{27})\)
Fixed field: Number field defined by a degree 27 polynomial

Values on generators

\(6\) → \(e\left(\frac{26}{27}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 109 }(9, a) \) \(1\)\(1\)\(e\left(\frac{8}{9}\right)\)\(e\left(\frac{2}{27}\right)\)\(e\left(\frac{7}{9}\right)\)\(e\left(\frac{5}{27}\right)\)\(e\left(\frac{26}{27}\right)\)\(e\left(\frac{14}{27}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{4}{27}\right)\)\(e\left(\frac{2}{27}\right)\)\(e\left(\frac{25}{27}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 109 }(9,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 109 }(9,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 109 }(9,·),\chi_{ 109 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 109 }(9,·)) \;\) at \(\; a,b = \) e.g. 1,2