Properties

Label 143.8
Modulus $143$
Conductor $143$
Order $20$
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(20))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([6,5]))
 
pari: [g,chi] = znchar(Mod(8,143))
 

Basic properties

Modulus: \(143\)
Conductor: \(143\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(20\)
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.s

\(\chi_{143}(8,\cdot)\) \(\chi_{143}(18,\cdot)\) \(\chi_{143}(57,\cdot)\) \(\chi_{143}(73,\cdot)\) \(\chi_{143}(83,\cdot)\) \(\chi_{143}(96,\cdot)\) \(\chi_{143}(112,\cdot)\) \(\chi_{143}(138,\cdot)\)

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: \(\Q(\zeta_{20})\)
Fixed field: 20.20.284589332775604260722209388186521117.1

Values on generators

\((79,67)\) → \((e\left(\frac{3}{10}\right),i)\)

Values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 143 }(8, a) \) \(1\)\(1\)\(e\left(\frac{11}{20}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{19}{20}\right)\)\(e\left(\frac{17}{20}\right)\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{4}{5}\right)\)\(1\)\(-1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 143 }(8,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 143 }(8,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 143 }(8,·),\chi_{ 143 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 143 }(8,·)) \;\) at \(\; a,b = \) e.g. 1,2