Properties

Label 143.20
Modulus 143143
Conductor 143143
Order 6060
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(143, base_ring=CyclotomicField(60)) M = H._module chi = DirichletCharacter(H, M([36,55]))
 
Copy content pari:[g,chi] = znchar(Mod(20,143))
 

Basic properties

Modulus: 143143
Conductor: 143143
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: 6060
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 143.x

χ143(15,)\chi_{143}(15,\cdot) χ143(20,)\chi_{143}(20,\cdot) χ143(37,)\chi_{143}(37,\cdot) χ143(58,)\chi_{143}(58,\cdot) χ143(59,)\chi_{143}(59,\cdot) χ143(71,)\chi_{143}(71,\cdot) χ143(80,)\chi_{143}(80,\cdot) χ143(93,)\chi_{143}(93,\cdot) χ143(97,)\chi_{143}(97,\cdot) χ143(102,)\chi_{143}(102,\cdot) χ143(115,)\chi_{143}(115,\cdot) χ143(119,)\chi_{143}(119,\cdot) χ143(124,)\chi_{143}(124,\cdot) χ143(136,)\chi_{143}(136,\cdot) χ143(137,)\chi_{143}(137,\cdot) χ143(141,)\chi_{143}(141,\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(ζ60)\Q(\zeta_{60})
Fixed field: Number field defined by a degree 60 polynomial

Values on generators

(79,67)(79,67)(e(35),e(1112))(e\left(\frac{3}{5}\right),e\left(\frac{11}{12}\right))

First values

aa 1-111223344556677889910101212
χ143(20,a) \chi_{ 143 }(20, a) 1-111e(3160)e\left(\frac{31}{60}\right)e(715)e\left(\frac{7}{15}\right)e(130)e\left(\frac{1}{30}\right)e(1320)e\left(\frac{13}{20}\right)e(5960)e\left(\frac{59}{60}\right)e(1760)e\left(\frac{17}{60}\right)e(1120)e\left(\frac{11}{20}\right)e(1415)e\left(\frac{14}{15}\right)e(16)e\left(\frac{1}{6}\right)1-1
Copy content sage:chi.jacobi_sum(n)
 
χ143(20,a)   \chi_{ 143 }(20,a) \; at   a=\;a = e.g. 2

Gauss sum

Copy content sage:chi.gauss_sum(a)
 
Copy content pari:znchargauss(g,chi,a)
 
τa(χ143(20,))   \tau_{ a }( \chi_{ 143 }(20,·) )\; at   a=\;a = e.g. 2

Jacobi sum

Copy content sage:chi.jacobi_sum(n)
 
J(χ143(20,),χ143(n,))   J(\chi_{ 143 }(20,·),\chi_{ 143 }(n,·)) \; for   n= \; n = e.g. 1

Kloosterman sum

Copy content sage:chi.kloosterman_sum(a,b)
 
K(a,b,χ143(20,))  K(a,b,\chi_{ 143 }(20,·)) \; at   a,b=\; a,b = e.g. 1,2