Properties

Label 475.122
Modulus 475475
Conductor 475475
Order 6060
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(60)) M = H._module chi = DirichletCharacter(H, M([51,10]))
 
Copy content pari:[g,chi] = znchar(Mod(122,475))
 

Basic properties

Modulus: 475475
Conductor: 475475
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: even
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Galois orbit 475.be

χ475(8,)\chi_{475}(8,\cdot) χ475(12,)\chi_{475}(12,\cdot) χ475(27,)\chi_{475}(27,\cdot) χ475(88,)\chi_{475}(88,\cdot) χ475(103,)\chi_{475}(103,\cdot) χ475(122,)\chi_{475}(122,\cdot) χ475(183,)\chi_{475}(183,\cdot) χ475(198,)\chi_{475}(198,\cdot) χ475(202,)\chi_{475}(202,\cdot) χ475(217,)\chi_{475}(217,\cdot) χ475(278,)\chi_{475}(278,\cdot) χ475(297,)\chi_{475}(297,\cdot) χ475(312,)\chi_{475}(312,\cdot) χ475(373,)\chi_{475}(373,\cdot) χ475(388,)\chi_{475}(388,\cdot) χ475(392,)\chi_{475}(392,\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

(77,401)(77,401)(e(1720),e(16))(e\left(\frac{17}{20}\right),e\left(\frac{1}{6}\right))

First values

aa 1-11122334466778899111112121313
χ475(122,a) \chi_{ 475 }(122, a) 1111e(160)e\left(\frac{1}{60}\right)e(760)e\left(\frac{7}{60}\right)e(130)e\left(\frac{1}{30}\right)e(215)e\left(\frac{2}{15}\right)iie(120)e\left(\frac{1}{20}\right)e(730)e\left(\frac{7}{30}\right)e(35)e\left(\frac{3}{5}\right)e(320)e\left(\frac{3}{20}\right)e(5960)e\left(\frac{59}{60}\right)
Copy content sage:chi.jacobi_sum(n)
 
χ475(122,a)   \chi_{ 475 }(122,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(122,))   \tau_{ a }( \chi_{ 475 }(122,·) )\; at   a=\;a = e.g. 2

Jacobi sum

Copy content sage:chi.jacobi_sum(n)
 
J(χ475(122,),χ475(n,))   J(\chi_{ 475 }(122,·),\chi_{ 475 }(n,·)) \; for   n= \; n = e.g. 1

Kloosterman sum

Copy content sage:chi.kloosterman_sum(a,b)
 
K(a,b,χ475(122,))  K(a,b,\chi_{ 475 }(122,·)) \; at   a,b=\; a,b = e.g. 1,2