Properties

Label 323.8
Modulus 323323
Conductor 323323
Order 2424
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(323, base_ring=CyclotomicField(24)) M = H._module chi = DirichletCharacter(H, M([15,4]))
 
Copy content pari:[g,chi] = znchar(Mod(8,323))
 

Basic properties

Modulus: 323323
Conductor: 323323
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: 2424
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 323.u

χ323(8,)\chi_{323}(8,\cdot) χ323(145,)\chi_{323}(145,\cdot) χ323(179,)\chi_{323}(179,\cdot) χ323(202,)\chi_{323}(202,\cdot) χ323(236,)\chi_{323}(236,\cdot) χ323(240,)\chi_{323}(240,\cdot) χ323(274,)\chi_{323}(274,\cdot) χ323(297,)\chi_{323}(297,\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(ζ24)\Q(\zeta_{24})
Fixed field: 24.0.2597163961920517723414089059021415099159693920192417.1

Values on generators

(20,154)(20,154)(e(58),e(16))(e\left(\frac{5}{8}\right),e\left(\frac{1}{6}\right))

First values

aa 1-111223344556677889910101111
χ323(8,a) \chi_{ 323 }(8, a) 1-111e(1112)e\left(\frac{11}{12}\right)e(1924)e\left(\frac{19}{24}\right)e(56)e\left(\frac{5}{6}\right)e(1924)e\left(\frac{19}{24}\right)e(1724)e\left(\frac{17}{24}\right)e(78)e\left(\frac{7}{8}\right)i-ie(712)e\left(\frac{7}{12}\right)e(1724)e\left(\frac{17}{24}\right)e(38)e\left(\frac{3}{8}\right)
Copy content sage:chi.jacobi_sum(n)
 
χ323(8,a)   \chi_{ 323 }(8,a) \; at   a=\;a = e.g. 2

Gauss sum

Copy content sage:chi.gauss_sum(a)
 
Copy content pari:znchargauss(g,chi,a)
 
τa(χ323(8,))   \tau_{ a }( \chi_{ 323 }(8,·) )\; at   a=\;a = e.g. 2

Jacobi sum

Copy content sage:chi.jacobi_sum(n)
 
J(χ323(8,),χ323(n,))   J(\chi_{ 323 }(8,·),\chi_{ 323 }(n,·)) \; for   n= \; n = e.g. 1

Kloosterman sum

Copy content sage:chi.kloosterman_sum(a,b)
 
K(a,b,χ323(8,))  K(a,b,\chi_{ 323 }(8,·)) \; at   a,b=\; a,b = e.g. 1,2