Properties

Label 143.114
Modulus $143$
Conductor $143$
Order $30$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(143, base_ring=CyclotomicField(30))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([6,25]))
 
pari: [g,chi] = znchar(Mod(114,143))
 

Basic properties

Modulus: \(143\)
Conductor: \(143\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(30\)
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 143.u

\(\chi_{143}(4,\cdot)\) \(\chi_{143}(36,\cdot)\) \(\chi_{143}(49,\cdot)\) \(\chi_{143}(69,\cdot)\) \(\chi_{143}(75,\cdot)\) \(\chi_{143}(82,\cdot)\) \(\chi_{143}(108,\cdot)\) \(\chi_{143}(114,\cdot)\)

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: \(\Q(\zeta_{15})\)
Fixed field: 30.30.69503752297329754905479727341904896738456941915804813.1

Values on generators

\((79,67)\) → \((e\left(\frac{1}{5}\right),e\left(\frac{5}{6}\right))\)

Values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 143 }(114, a) \) \(1\)\(1\)\(e\left(\frac{1}{30}\right)\)\(e\left(\frac{14}{15}\right)\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{29}{30}\right)\)\(e\left(\frac{17}{30}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{13}{15}\right)\)\(e\left(\frac{1}{3}\right)\)\(1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 143 }(114,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 143 }(114,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 143 }(114,·),\chi_{ 143 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 143 }(114,·)) \;\) at \(\; a,b = \) e.g. 1,2