Properties

Label 289.69
Modulus $289$
Conductor $289$
Order $17$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(289, base_ring=CyclotomicField(34))
 
M = H._module
 
chi = DirichletCharacter(H, M([28]))
 
pari: [g,chi] = znchar(Mod(69,289))
 

Basic properties

Modulus: \(289\)
Conductor: \(289\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(17\)
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 289.f

\(\chi_{289}(18,\cdot)\) \(\chi_{289}(35,\cdot)\) \(\chi_{289}(52,\cdot)\) \(\chi_{289}(69,\cdot)\) \(\chi_{289}(86,\cdot)\) \(\chi_{289}(103,\cdot)\) \(\chi_{289}(120,\cdot)\) \(\chi_{289}(137,\cdot)\) \(\chi_{289}(154,\cdot)\) \(\chi_{289}(171,\cdot)\) \(\chi_{289}(188,\cdot)\) \(\chi_{289}(205,\cdot)\) \(\chi_{289}(222,\cdot)\) \(\chi_{289}(239,\cdot)\) \(\chi_{289}(256,\cdot)\) \(\chi_{289}(273,\cdot)\)

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: \(\Q(\zeta_{17})\)
Fixed field: Number field defined by a degree 17 polynomial

Values on generators

\(3\) → \(e\left(\frac{14}{17}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 289 }(69, a) \) \(1\)\(1\)\(e\left(\frac{8}{17}\right)\)\(e\left(\frac{14}{17}\right)\)\(e\left(\frac{16}{17}\right)\)\(e\left(\frac{10}{17}\right)\)\(e\left(\frac{5}{17}\right)\)\(e\left(\frac{11}{17}\right)\)\(e\left(\frac{7}{17}\right)\)\(e\left(\frac{11}{17}\right)\)\(e\left(\frac{1}{17}\right)\)\(e\left(\frac{16}{17}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 289 }(69,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 289 }(69,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 289 }(69,·),\chi_{ 289 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 289 }(69,·)) \;\) at \(\; a,b = \) e.g. 1,2