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Properties

Label 867.152
Modulus $867$
Conductor $867$
Order $34$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(867, base_ring=CyclotomicField(34)) M = H._module chi = DirichletCharacter(H, M([17,5]))
 
Copy content pari:[g,chi] = znchar(Mod(152,867))
 

Basic properties

Modulus: \(867\)
Conductor: \(867\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(34\)
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 867.m

\(\chi_{867}(50,\cdot)\) \(\chi_{867}(101,\cdot)\) \(\chi_{867}(152,\cdot)\) \(\chi_{867}(203,\cdot)\) \(\chi_{867}(254,\cdot)\) \(\chi_{867}(305,\cdot)\) \(\chi_{867}(356,\cdot)\) \(\chi_{867}(407,\cdot)\) \(\chi_{867}(458,\cdot)\) \(\chi_{867}(509,\cdot)\) \(\chi_{867}(560,\cdot)\) \(\chi_{867}(611,\cdot)\) \(\chi_{867}(662,\cdot)\) \(\chi_{867}(713,\cdot)\) \(\chi_{867}(764,\cdot)\) \(\chi_{867}(815,\cdot)\)

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: \(\Q(\zeta_{17})\)
Fixed field: 34.0.12309522877752896848740501655027735029115386449381382152715887725819844838409975062613491.1

Values on generators

\((290,292)\) → \((-1,e\left(\frac{5}{34}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 867 }(152, a) \) \(-1\)\(1\)\(e\left(\frac{15}{34}\right)\)\(e\left(\frac{15}{17}\right)\)\(e\left(\frac{3}{17}\right)\)\(e\left(\frac{27}{34}\right)\)\(e\left(\frac{11}{34}\right)\)\(e\left(\frac{21}{34}\right)\)\(e\left(\frac{15}{17}\right)\)\(e\left(\frac{14}{17}\right)\)\(e\left(\frac{4}{17}\right)\)\(e\left(\frac{13}{17}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 867 }(152,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

Copy content sage:chi.gauss_sum(a)
 
Copy content pari:znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 867 }(152,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

Copy content sage:chi.jacobi_sum(n)
 
\( J(\chi_{ 867 }(152,·),\chi_{ 867 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

Copy content sage:chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 867 }(152,·)) \;\) at \(\; a,b = \) e.g. 1,2