Properties

Label 93.50
Modulus 9393
Conductor 9393
Order 3030
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(93, base_ring=CyclotomicField(30)) M = H._module chi = DirichletCharacter(H, M([15,4]))
 
Copy content pari:[g,chi] = znchar(Mod(50,93))
 

Basic properties

Modulus: 9393
Conductor: 9393
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: 3030
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 93.o

χ93(14,)\chi_{93}(14,\cdot) χ93(20,)\chi_{93}(20,\cdot) χ93(38,)\chi_{93}(38,\cdot) χ93(41,)\chi_{93}(41,\cdot) χ93(50,)\chi_{93}(50,\cdot) χ93(59,)\chi_{93}(59,\cdot) χ93(71,)\chi_{93}(71,\cdot) χ93(80,)\chi_{93}(80,\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(ζ15)\Q(\zeta_{15})
Fixed field: 30.0.8221408887534945302579972677458642036796803806187.1

Values on generators

(32,34)(32,34)(1,e(215))(-1,e\left(\frac{2}{15}\right))

First values

aa 1-111224455778810101111131314141616
χ93(50,a) \chi_{ 93 }(50, a) 1-111e(710)e\left(\frac{7}{10}\right)e(25)e\left(\frac{2}{5}\right)e(16)e\left(\frac{1}{6}\right)e(1115)e\left(\frac{11}{15}\right)e(110)e\left(\frac{1}{10}\right)e(1315)e\left(\frac{13}{15}\right)e(1730)e\left(\frac{17}{30}\right)e(715)e\left(\frac{7}{15}\right)e(1330)e\left(\frac{13}{30}\right)e(45)e\left(\frac{4}{5}\right)
Copy content sage:chi.jacobi_sum(n)
 
χ93(50,a)   \chi_{ 93 }(50,a) \; at   a=\;a = e.g. 2

Gauss sum

Copy content sage:chi.gauss_sum(a)
 
Copy content pari:znchargauss(g,chi,a)
 
τa(χ93(50,))   \tau_{ a }( \chi_{ 93 }(50,·) )\; at   a=\;a = e.g. 2

Jacobi sum

Copy content sage:chi.jacobi_sum(n)
 
J(χ93(50,),χ93(n,))   J(\chi_{ 93 }(50,·),\chi_{ 93 }(n,·)) \; for   n= \; n = e.g. 1

Kloosterman sum

Copy content sage:chi.kloosterman_sum(a,b)
 
K(a,b,χ93(50,))  K(a,b,\chi_{ 93 }(50,·)) \; at   a,b=\; a,b = e.g. 1,2