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Properties

Label 520.419
Modulus 520520
Conductor 520520
Order 66
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(520, base_ring=CyclotomicField(6))
 
M = H._module
 
chi = DirichletCharacter(H, M([3,3,3,2]))
 
pari: [g,chi] = znchar(Mod(419,520))
 

Basic properties

Modulus: 520520
Conductor: 520520
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: 66
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 520.bx

χ520(139,⋅)\chi_{520}(139,\cdot) χ520(419,⋅)\chi_{520}(419,\cdot)

sage: chi.galois_orbit()
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: Q(ζ3)\mathbb{Q}(\zeta_3)
Fixed field: 6.0.1827904000.2

Values on generators

(391,261,417,41)(391,261,417,41) → (−1,−1,−1,e(13))(-1,-1,-1,e\left(\frac{1}{3}\right))

First values

aa −1-1113377991111171719192121232327272929
χ520(419,a) \chi_{ 520 }(419, a) −1-111e(56)e\left(\frac{5}{6}\right)e(23)e\left(\frac{2}{3}\right)e(23)e\left(\frac{2}{3}\right)e(13)e\left(\frac{1}{3}\right)e(16)e\left(\frac{1}{6}\right)e(23)e\left(\frac{2}{3}\right)−1-1e(13)e\left(\frac{1}{3}\right)−1-1e(56)e\left(\frac{5}{6}\right)
sage: chi.jacobi_sum(n)
 
χ520(419,a)   \chi_{ 520 }(419,a) \; at   a=\;a = e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
τa(χ520(419,⋅))   \tau_{ a }( \chi_{ 520 }(419,·) )\; at   a=\;a = e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
J(χ520(419,⋅),χ520(n,⋅))   J(\chi_{ 520 }(419,·),\chi_{ 520 }(n,·)) \; for   n= \; n = e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
K(a,b,χ520(419,⋅))  K(a,b,\chi_{ 520 }(419,·)) \; at   a,b=\; a,b = e.g. 1,2