Properties

Label 675.16
Modulus 675675
Conductor 675675
Order 4545
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(675, base_ring=CyclotomicField(90)) M = H._module chi = DirichletCharacter(H, M([20,18]))
 
Copy content pari:[g,chi] = znchar(Mod(16,675))
 

Basic properties

Modulus: 675675
Conductor: 675675
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: 4545
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 675.bc

χ675(16,)\chi_{675}(16,\cdot) χ675(31,)\chi_{675}(31,\cdot) χ675(61,)\chi_{675}(61,\cdot) χ675(106,)\chi_{675}(106,\cdot) χ675(121,)\chi_{675}(121,\cdot) χ675(166,)\chi_{675}(166,\cdot) χ675(196,)\chi_{675}(196,\cdot) χ675(211,)\chi_{675}(211,\cdot) χ675(241,)\chi_{675}(241,\cdot) χ675(256,)\chi_{675}(256,\cdot) χ675(286,)\chi_{675}(286,\cdot) χ675(331,)\chi_{675}(331,\cdot) χ675(346,)\chi_{675}(346,\cdot) χ675(391,)\chi_{675}(391,\cdot) χ675(421,)\chi_{675}(421,\cdot) χ675(436,)\chi_{675}(436,\cdot) χ675(466,)\chi_{675}(466,\cdot) χ675(481,)\chi_{675}(481,\cdot) χ675(511,)\chi_{675}(511,\cdot) χ675(556,)\chi_{675}(556,\cdot) χ675(571,)\chi_{675}(571,\cdot) χ675(616,)\chi_{675}(616,\cdot) χ675(646,)\chi_{675}(646,\cdot) χ675(661,)\chi_{675}(661,\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 45 polynomial

Values on generators

(326,352)(326,352)(e(29),e(15))(e\left(\frac{2}{9}\right),e\left(\frac{1}{5}\right))

First values

aa 1-11122447788111113131414161617171919
χ675(16,a) \chi_{ 675 }(16, a) 1111e(1945)e\left(\frac{19}{45}\right)e(3845)e\left(\frac{38}{45}\right)e(59)e\left(\frac{5}{9}\right)e(415)e\left(\frac{4}{15}\right)e(445)e\left(\frac{4}{45}\right)e(2645)e\left(\frac{26}{45}\right)e(4445)e\left(\frac{44}{45}\right)e(3145)e\left(\frac{31}{45}\right)e(1415)e\left(\frac{14}{15}\right)e(415)e\left(\frac{4}{15}\right)
Copy content sage:chi.jacobi_sum(n)
 
χ675(16,a)   \chi_{ 675 }(16,a) \; at   a=\;a = e.g. 2

Gauss sum

Copy content sage:chi.gauss_sum(a)
 
Copy content pari:znchargauss(g,chi,a)
 
τa(χ675(16,))   \tau_{ a }( \chi_{ 675 }(16,·) )\; at   a=\;a = e.g. 2

Jacobi sum

Copy content sage:chi.jacobi_sum(n)
 
J(χ675(16,),χ675(n,))   J(\chi_{ 675 }(16,·),\chi_{ 675 }(n,·)) \; for   n= \; n = e.g. 1

Kloosterman sum

Copy content sage:chi.kloosterman_sum(a,b)
 
K(a,b,χ675(16,))  K(a,b,\chi_{ 675 }(16,·)) \; at   a,b=\; a,b = e.g. 1,2