Properties

Label 585.437
Modulus 585585
Conductor 585585
Order 1212
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Downloads

Learn more

Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(585, base_ring=CyclotomicField(12)) M = H._module chi = DirichletCharacter(H, M([10,3,3]))
 
Copy content pari:[g,chi] = znchar(Mod(437,585))
 

Basic properties

Modulus: 585585
Conductor: 585585
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: 1212
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 585.cd

χ585(47,)\chi_{585}(47,\cdot) χ585(83,)\chi_{585}(83,\cdot) χ585(437,)\chi_{585}(437,\cdot) χ585(473,)\chi_{585}(473,\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(ζ12)\Q(\zeta_{12})
Fixed field: 12.0.8024219399780963666015625.1

Values on generators

(326,352,496)(326,352,496)(e(56),i,i)(e\left(\frac{5}{6}\right),i,i)

First values

aa 1-11122447788111114141616171719192222
χ585(437,a) \chi_{ 585 }(437, a) 1-111e(13)e\left(\frac{1}{3}\right)e(23)e\left(\frac{2}{3}\right)e(13)e\left(\frac{1}{3}\right)11e(712)e\left(\frac{7}{12}\right)e(23)e\left(\frac{2}{3}\right)e(13)e\left(\frac{1}{3}\right)iii-ie(1112)e\left(\frac{11}{12}\right)
Copy content sage:chi.jacobi_sum(n)
 
χ585(437,a)   \chi_{ 585 }(437,a) \; at   a=\;a = e.g. 2

Gauss sum

Copy content sage:chi.gauss_sum(a)
 
Copy content pari:znchargauss(g,chi,a)
 
τa(χ585(437,))   \tau_{ a }( \chi_{ 585 }(437,·) )\; at   a=\;a = e.g. 2

Jacobi sum

Copy content sage:chi.jacobi_sum(n)
 
J(χ585(437,),χ585(n,))   J(\chi_{ 585 }(437,·),\chi_{ 585 }(n,·)) \; for   n= \; n = e.g. 1

Kloosterman sum

Copy content sage:chi.kloosterman_sum(a,b)
 
K(a,b,χ585(437,))  K(a,b,\chi_{ 585 }(437,·)) \; at   a,b=\; a,b = e.g. 1,2