Properties

Label 109.18
Modulus $109$
Conductor $109$
Order $108$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(109, base_ring=CyclotomicField(108))
 
M = H._module
 
chi = DirichletCharacter(H, M([53]))
 
pari: [g,chi] = znchar(Mod(18,109))
 

Basic properties

Modulus: \(109\)
Conductor: \(109\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(108\)
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 109.l

\(\chi_{109}(6,\cdot)\) \(\chi_{109}(10,\cdot)\) \(\chi_{109}(11,\cdot)\) \(\chi_{109}(13,\cdot)\) \(\chi_{109}(14,\cdot)\) \(\chi_{109}(18,\cdot)\) \(\chi_{109}(24,\cdot)\) \(\chi_{109}(30,\cdot)\) \(\chi_{109}(37,\cdot)\) \(\chi_{109}(39,\cdot)\) \(\chi_{109}(40,\cdot)\) \(\chi_{109}(42,\cdot)\) \(\chi_{109}(44,\cdot)\) \(\chi_{109}(47,\cdot)\) \(\chi_{109}(50,\cdot)\) \(\chi_{109}(51,\cdot)\) \(\chi_{109}(52,\cdot)\) \(\chi_{109}(53,\cdot)\) \(\chi_{109}(56,\cdot)\) \(\chi_{109}(57,\cdot)\) \(\chi_{109}(58,\cdot)\) \(\chi_{109}(59,\cdot)\) \(\chi_{109}(62,\cdot)\) \(\chi_{109}(65,\cdot)\) \(\chi_{109}(67,\cdot)\) \(\chi_{109}(69,\cdot)\) \(\chi_{109}(70,\cdot)\) \(\chi_{109}(72,\cdot)\) \(\chi_{109}(79,\cdot)\) \(\chi_{109}(85,\cdot)\) ...

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: $\Q(\zeta_{108})$
Fixed field: Number field defined by a degree 108 polynomial (not computed)

Values on generators

\(6\) → \(e\left(\frac{53}{108}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 109 }(18, a) \) \(-1\)\(1\)\(e\left(\frac{35}{36}\right)\)\(e\left(\frac{14}{27}\right)\)\(e\left(\frac{17}{18}\right)\)\(e\left(\frac{8}{27}\right)\)\(e\left(\frac{53}{108}\right)\)\(e\left(\frac{17}{27}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{1}{27}\right)\)\(e\left(\frac{29}{108}\right)\)\(e\left(\frac{79}{108}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 109 }(18,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 109 }(18,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 109 }(18,·),\chi_{ 109 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 109 }(18,·)) \;\) at \(\; a,b = \) e.g. 1,2