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Properties

Label 169.21
Modulus $169$
Conductor $169$
Order $52$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(169, base_ring=CyclotomicField(52)) M = H._module chi = DirichletCharacter(H, M([25]))
 
Copy content pari:[g,chi] = znchar(Mod(21,169))
 

Basic properties

Modulus: \(169\)
Conductor: \(169\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(52\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: yes
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Galois orbit 169.j

\(\chi_{169}(5,\cdot)\) \(\chi_{169}(8,\cdot)\) \(\chi_{169}(18,\cdot)\) \(\chi_{169}(21,\cdot)\) \(\chi_{169}(31,\cdot)\) \(\chi_{169}(34,\cdot)\) \(\chi_{169}(44,\cdot)\) \(\chi_{169}(47,\cdot)\) \(\chi_{169}(57,\cdot)\) \(\chi_{169}(60,\cdot)\) \(\chi_{169}(73,\cdot)\) \(\chi_{169}(83,\cdot)\) \(\chi_{169}(86,\cdot)\) \(\chi_{169}(96,\cdot)\) \(\chi_{169}(109,\cdot)\) \(\chi_{169}(112,\cdot)\) \(\chi_{169}(122,\cdot)\) \(\chi_{169}(125,\cdot)\) \(\chi_{169}(135,\cdot)\) \(\chi_{169}(138,\cdot)\) \(\chi_{169}(148,\cdot)\) \(\chi_{169}(151,\cdot)\) \(\chi_{169}(161,\cdot)\) \(\chi_{169}(164,\cdot)\)

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: $\Q(\zeta_{52})$
Fixed field: Number field defined by a degree 52 polynomial

Values on generators

\(2\) → \(e\left(\frac{25}{52}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 169 }(21, a) \) \(-1\)\(1\)\(e\left(\frac{25}{52}\right)\)\(e\left(\frac{8}{13}\right)\)\(e\left(\frac{25}{26}\right)\)\(e\left(\frac{17}{52}\right)\)\(e\left(\frac{5}{52}\right)\)\(e\left(\frac{23}{52}\right)\)\(e\left(\frac{23}{52}\right)\)\(e\left(\frac{3}{13}\right)\)\(e\left(\frac{21}{26}\right)\)\(e\left(\frac{27}{52}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 169 }(21,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

Copy content sage:chi.gauss_sum(a)
 
Copy content pari:znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 169 }(21,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

Copy content sage:chi.jacobi_sum(n)
 
\( J(\chi_{ 169 }(21,·),\chi_{ 169 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

Copy content sage:chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 169 }(21,·)) \;\) at \(\; a,b = \) e.g. 1,2