Properties

Label 475.119
Modulus 475475
Conductor 475475
Order 9090
Real no
Primitive yes
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(475, base_ring=CyclotomicField(90)) M = H._module chi = DirichletCharacter(H, M([81,80]))
 
Copy content pari:[g,chi] = znchar(Mod(119,475))
 

Basic properties

Modulus: 475475
Conductor: 475475
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: 9090
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 475.bg

χ475(4,)\chi_{475}(4,\cdot) χ475(9,)\chi_{475}(9,\cdot) χ475(44,)\chi_{475}(44,\cdot) χ475(54,)\chi_{475}(54,\cdot) χ475(104,)\chi_{475}(104,\cdot) χ475(119,)\chi_{475}(119,\cdot) χ475(139,)\chi_{475}(139,\cdot) χ475(169,)\chi_{475}(169,\cdot) χ475(194,)\chi_{475}(194,\cdot) χ475(214,)\chi_{475}(214,\cdot) χ475(234,)\chi_{475}(234,\cdot) χ475(244,)\chi_{475}(244,\cdot) χ475(264,)\chi_{475}(264,\cdot) χ475(289,)\chi_{475}(289,\cdot) χ475(294,)\chi_{475}(294,\cdot) χ475(309,)\chi_{475}(309,\cdot) χ475(329,)\chi_{475}(329,\cdot) χ475(339,)\chi_{475}(339,\cdot) χ475(359,)\chi_{475}(359,\cdot) χ475(384,)\chi_{475}(384,\cdot) χ475(389,)\chi_{475}(389,\cdot) χ475(404,)\chi_{475}(404,\cdot) χ475(434,)\chi_{475}(434,\cdot) χ475(454,)\chi_{475}(454,\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(ζ45)\Q(\zeta_{45})
Fixed field: Number field defined by a degree 90 polynomial

Values on generators

(77,401)(77,401)(e(910),e(89))(e\left(\frac{9}{10}\right),e\left(\frac{8}{9}\right))

First values

aa 1-11122334466778899111112121313
χ475(119,a) \chi_{ 475 }(119, a) 1111e(7190)e\left(\frac{71}{90}\right)e(7790)e\left(\frac{77}{90}\right)e(2645)e\left(\frac{26}{45}\right)e(2945)e\left(\frac{29}{45}\right)e(56)e\left(\frac{5}{6}\right)e(1130)e\left(\frac{11}{30}\right)e(3245)e\left(\frac{32}{45}\right)e(115)e\left(\frac{1}{15}\right)e(1330)e\left(\frac{13}{30}\right)e(4990)e\left(\frac{49}{90}\right)
Copy content sage:chi.jacobi_sum(n)
 
χ475(119,a)   \chi_{ 475 }(119,a) \; at   a=\;a = e.g. 2

Gauss sum

Copy content sage:chi.gauss_sum(a)
 
Copy content pari:znchargauss(g,chi,a)
 
τa(χ475(119,))   \tau_{ a }( \chi_{ 475 }(119,·) )\; at   a=\;a = e.g. 2

Jacobi sum

Copy content sage:chi.jacobi_sum(n)
 
J(χ475(119,),χ475(n,))   J(\chi_{ 475 }(119,·),\chi_{ 475 }(n,·)) \; for   n= \; n = e.g. 1

Kloosterman sum

Copy content sage:chi.kloosterman_sum(a,b)
 
K(a,b,χ475(119,))  K(a,b,\chi_{ 475 }(119,·)) \; at   a,b=\; a,b = e.g. 1,2