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Properties

Label 875.869
Modulus $875$
Conductor $125$
Order $50$
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(875, base_ring=CyclotomicField(50)) M = H._module chi = DirichletCharacter(H, M([29,0]))
 
Copy content pari:[g,chi] = znchar(Mod(869,875))
 

Basic properties

Modulus: \(875\)
Conductor: \(125\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(50\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{125}(119,\cdot)\)
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 875.y

\(\chi_{875}(29,\cdot)\) \(\chi_{875}(64,\cdot)\) \(\chi_{875}(134,\cdot)\) \(\chi_{875}(169,\cdot)\) \(\chi_{875}(204,\cdot)\) \(\chi_{875}(239,\cdot)\) \(\chi_{875}(309,\cdot)\) \(\chi_{875}(344,\cdot)\) \(\chi_{875}(379,\cdot)\) \(\chi_{875}(414,\cdot)\) \(\chi_{875}(484,\cdot)\) \(\chi_{875}(519,\cdot)\) \(\chi_{875}(554,\cdot)\) \(\chi_{875}(589,\cdot)\) \(\chi_{875}(659,\cdot)\) \(\chi_{875}(694,\cdot)\) \(\chi_{875}(729,\cdot)\) \(\chi_{875}(764,\cdot)\) \(\chi_{875}(834,\cdot)\) \(\chi_{875}(869,\cdot)\)

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: \(\Q(\zeta_{25})\)
Fixed field: Number field defined by a degree 50 polynomial

Values on generators

\((127,626)\) → \((e\left(\frac{29}{50}\right),1)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(11\)\(12\)\(13\)\(16\)
\( \chi_{ 875 }(869, a) \) \(1\)\(1\)\(e\left(\frac{29}{50}\right)\)\(e\left(\frac{3}{50}\right)\)\(e\left(\frac{4}{25}\right)\)\(e\left(\frac{16}{25}\right)\)\(e\left(\frac{37}{50}\right)\)\(e\left(\frac{3}{25}\right)\)\(e\left(\frac{2}{25}\right)\)\(e\left(\frac{11}{50}\right)\)\(e\left(\frac{31}{50}\right)\)\(e\left(\frac{8}{25}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 875 }(869,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

Copy content sage:chi.gauss_sum(a)
 
Copy content pari:znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 875 }(869,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

Copy content sage:chi.jacobi_sum(n)
 
\( J(\chi_{ 875 }(869,·),\chi_{ 875 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

Copy content sage:chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 875 }(869,·)) \;\) at \(\; a,b = \) e.g. 1,2