Properties

Label 99.47
Modulus 9999
Conductor 9999
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(99, base_ring=CyclotomicField(30)) M = H._module chi = DirichletCharacter(H, M([5,24]))
 
Copy content pari:[g,chi] = znchar(Mod(47,99))
 

Basic properties

Modulus: 9999
Conductor: 9999
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 99.n

χ99(5,)\chi_{99}(5,\cdot) χ99(14,)\chi_{99}(14,\cdot) χ99(20,)\chi_{99}(20,\cdot) χ99(38,)\chi_{99}(38,\cdot) χ99(47,)\chi_{99}(47,\cdot) χ99(59,)\chi_{99}(59,\cdot) χ99(86,)\chi_{99}(86,\cdot) χ99(92,)\chi_{99}(92,\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: 30.0.29099190400267368949073680941341991556317731763.1

Values on generators

(56,46)(56,46)(e(16),e(45))(e\left(\frac{1}{6}\right),e\left(\frac{4}{5}\right))

First values

aa 1-111224455778810101313141416161717
χ99(47,a) \chi_{ 99 }(47, a) 1-111e(2930)e\left(\frac{29}{30}\right)e(1415)e\left(\frac{14}{15}\right)e(130)e\left(\frac{1}{30}\right)e(415)e\left(\frac{4}{15}\right)e(910)e\left(\frac{9}{10}\right)11e(215)e\left(\frac{2}{15}\right)e(730)e\left(\frac{7}{30}\right)e(1315)e\left(\frac{13}{15}\right)e(710)e\left(\frac{7}{10}\right)
Copy content sage:chi.jacobi_sum(n)
 
χ99(47,a)   \chi_{ 99 }(47,a) \; at   a=\;a = e.g. 2

Gauss sum

Copy content sage:chi.gauss_sum(a)
 
Copy content pari:znchargauss(g,chi,a)
 
τa(χ99(47,))   \tau_{ a }( \chi_{ 99 }(47,·) )\; at   a=\;a = e.g. 2

Jacobi sum

Copy content sage:chi.jacobi_sum(n)
 
J(χ99(47,),χ99(n,))   J(\chi_{ 99 }(47,·),\chi_{ 99 }(n,·)) \; for   n= \; n = e.g. 1

Kloosterman sum

Copy content sage:chi.kloosterman_sum(a,b)
 
K(a,b,χ99(47,))  K(a,b,\chi_{ 99 }(47,·)) \; at   a,b=\; a,b = e.g. 1,2