Properties

Label 239.232
Modulus $239$
Conductor $239$
Order $119$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(239, base_ring=CyclotomicField(238))
 
M = H._module
 
chi = DirichletCharacter(H, M([120]))
 
pari: [g,chi] = znchar(Mod(232,239))
 

Basic properties

Modulus: \(239\)
Conductor: \(239\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(119\)
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 239.g

\(\chi_{239}(2,\cdot)\) \(\chi_{239}(3,\cdot)\) \(\chi_{239}(4,\cdot)\) \(\chi_{239}(5,\cdot)\) \(\chi_{239}(8,\cdot)\) \(\chi_{239}(9,\cdot)\) \(\chi_{239}(11,\cdot)\) \(\chi_{239}(12,\cdot)\) \(\chi_{239}(15,\cdot)\) \(\chi_{239}(16,\cdot)\) \(\chi_{239}(17,\cdot)\) \(\chi_{239}(18,\cdot)\) \(\chi_{239}(20,\cdot)\) \(\chi_{239}(25,\cdot)\) \(\chi_{239}(27,\cdot)\) \(\chi_{239}(29,\cdot)\) \(\chi_{239}(30,\cdot)\) \(\chi_{239}(31,\cdot)\) \(\chi_{239}(32,\cdot)\) \(\chi_{239}(33,\cdot)\) \(\chi_{239}(34,\cdot)\) \(\chi_{239}(45,\cdot)\) \(\chi_{239}(48,\cdot)\) \(\chi_{239}(49,\cdot)\) \(\chi_{239}(50,\cdot)\) \(\chi_{239}(54,\cdot)\) \(\chi_{239}(55,\cdot)\) \(\chi_{239}(58,\cdot)\) \(\chi_{239}(60,\cdot)\) \(\chi_{239}(61,\cdot)\) ...

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: $\Q(\zeta_{119})$
Fixed field: Number field defined by a degree 119 polynomial (not computed)

Values on generators

\(7\) → \(e\left(\frac{60}{119}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 239 }(232, a) \) \(1\)\(1\)\(e\left(\frac{33}{119}\right)\)\(e\left(\frac{37}{119}\right)\)\(e\left(\frac{66}{119}\right)\)\(e\left(\frac{69}{119}\right)\)\(e\left(\frac{10}{17}\right)\)\(e\left(\frac{60}{119}\right)\)\(e\left(\frac{99}{119}\right)\)\(e\left(\frac{74}{119}\right)\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{2}{119}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 239 }(232,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 239 }(232,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 239 }(232,·),\chi_{ 239 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 239 }(232,·)) \;\) at \(\; a,b = \) e.g. 1,2