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Properties

Label 158.21
Modulus $158$
Conductor $79$
Order $13$
Real no
Primitive no
Minimal yes
Parity even

Related objects

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(158, base_ring=CyclotomicField(26)) M = H._module chi = DirichletCharacter(H, M([18]))
 
Copy content pari:[g,chi] = znchar(Mod(21,158))
 

Basic properties

Modulus: \(158\)
Conductor: \(79\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(13\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{79}(21,\cdot)\)
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Galois orbit 158.e

\(\chi_{158}(21,\cdot)\) \(\chi_{158}(65,\cdot)\) \(\chi_{158}(67,\cdot)\) \(\chi_{158}(87,\cdot)\) \(\chi_{158}(89,\cdot)\) \(\chi_{158}(97,\cdot)\) \(\chi_{158}(101,\cdot)\) \(\chi_{158}(117,\cdot)\) \(\chi_{158}(125,\cdot)\) \(\chi_{158}(131,\cdot)\) \(\chi_{158}(141,\cdot)\) \(\chi_{158}(143,\cdot)\)

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

Related number fields

Field of values: \(\Q(\zeta_{13})\)
Fixed field: Number field defined by a degree 13 polynomial

Values on generators

\(3\) → \(e\left(\frac{9}{13}\right)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 158 }(21, a) \) \(1\)\(1\)\(e\left(\frac{9}{13}\right)\)\(e\left(\frac{12}{13}\right)\)\(e\left(\frac{9}{13}\right)\)\(e\left(\frac{5}{13}\right)\)\(e\left(\frac{1}{13}\right)\)\(e\left(\frac{7}{13}\right)\)\(e\left(\frac{8}{13}\right)\)\(e\left(\frac{7}{13}\right)\)\(e\left(\frac{2}{13}\right)\)\(e\left(\frac{5}{13}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 158 }(21,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

Copy content sage:chi.gauss_sum(a)
 
Copy content pari:znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 158 }(21,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

Copy content sage:chi.jacobi_sum(n)
 
\( J(\chi_{ 158 }(21,·),\chi_{ 158 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

Copy content sage:chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 158 }(21,·)) \;\) at \(\; a,b = \) e.g. 1,2