Properties

Label 173.95
Modulus 173173
Conductor 173173
Order 4343
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(173, base_ring=CyclotomicField(86)) M = H._module chi = DirichletCharacter(H, M([36]))
 
Copy content pari:[g,chi] = znchar(Mod(95,173))
 

Basic properties

Modulus: 173173
Conductor: 173173
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: 4343
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: even
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Galois orbit 173.d

χ173(6,)\chi_{173}(6,\cdot) χ173(10,)\chi_{173}(10,\cdot) χ173(14,)\chi_{173}(14,\cdot) χ173(16,)\chi_{173}(16,\cdot) χ173(22,)\chi_{173}(22,\cdot) χ173(23,)\chi_{173}(23,\cdot) χ173(29,)\chi_{173}(29,\cdot) χ173(36,)\chi_{173}(36,\cdot) χ173(43,)\chi_{173}(43,\cdot) χ173(47,)\chi_{173}(47,\cdot) χ173(51,)\chi_{173}(51,\cdot) χ173(52,)\chi_{173}(52,\cdot) χ173(57,)\chi_{173}(57,\cdot) χ173(60,)\chi_{173}(60,\cdot) χ173(81,)\chi_{173}(81,\cdot) χ173(83,)\chi_{173}(83,\cdot) χ173(84,)\chi_{173}(84,\cdot) χ173(85,)\chi_{173}(85,\cdot) χ173(95,)\chi_{173}(95,\cdot) χ173(96,)\chi_{173}(96,\cdot) χ173(100,)\chi_{173}(100,\cdot) χ173(106,)\chi_{173}(106,\cdot) χ173(109,)\chi_{173}(109,\cdot) χ173(117,)\chi_{173}(117,\cdot) χ173(118,)\chi_{173}(118,\cdot) χ173(119,)\chi_{173}(119,\cdot) χ173(124,)\chi_{173}(124,\cdot) χ173(132,)\chi_{173}(132,\cdot) χ173(133,)\chi_{173}(133,\cdot) χ173(135,)\chi_{173}(135,\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(ζ43)\Q(\zeta_{43})
Fixed field: Number field defined by a degree 43 polynomial

Values on generators

22e(1843)e\left(\frac{18}{43}\right)

First values

aa 1-111223344556677889910101111
χ173(95,a) \chi_{ 173 }(95, a) 1111e(1843)e\left(\frac{18}{43}\right)e(1343)e\left(\frac{13}{43}\right)e(3643)e\left(\frac{36}{43}\right)e(1443)e\left(\frac{14}{43}\right)e(3143)e\left(\frac{31}{43}\right)e(3343)e\left(\frac{33}{43}\right)e(1143)e\left(\frac{11}{43}\right)e(2643)e\left(\frac{26}{43}\right)e(3243)e\left(\frac{32}{43}\right)e(2743)e\left(\frac{27}{43}\right)
Copy content sage:chi.jacobi_sum(n)
 
χ173(95,a)   \chi_{ 173 }(95,a) \; at   a=\;a = e.g. 2

Gauss sum

Copy content sage:chi.gauss_sum(a)
 
Copy content pari:znchargauss(g,chi,a)
 
τa(χ173(95,))   \tau_{ a }( \chi_{ 173 }(95,·) )\; at   a=\;a = e.g. 2

Jacobi sum

Copy content sage:chi.jacobi_sum(n)
 
J(χ173(95,),χ173(n,))   J(\chi_{ 173 }(95,·),\chi_{ 173 }(n,·)) \; for   n= \; n = e.g. 1

Kloosterman sum

Copy content sage:chi.kloosterman_sum(a,b)
 
K(a,b,χ173(95,))  K(a,b,\chi_{ 173 }(95,·)) \; at   a,b=\; a,b = e.g. 1,2