Properties

Label 851.484
Modulus 851851
Conductor 3737
Order 1818
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(851, base_ring=CyclotomicField(18))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,13]))
 
pari: [g,chi] = znchar(Mod(484,851))
 

Basic properties

Modulus: 851851
Conductor: 3737
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: 1818
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from χ37(3,)\chi_{37}(3,\cdot)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 851.o

χ851(139,)\chi_{851}(139,\cdot) χ851(300,)\chi_{851}(300,\cdot) χ851(484,)\chi_{851}(484,\cdot) χ851(576,)\chi_{851}(576,\cdot) χ851(622,)\chi_{851}(622,\cdot) χ851(691,)\chi_{851}(691,\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(ζ9)\Q(\zeta_{9})
Fixed field: Number field defined by a degree 18 polynomial

Values on generators

(741,668)(741,668)(1,e(1318))(1,e\left(\frac{13}{18}\right))

First values

aa 1-111223344556677889910101111
χ851(484,a) \chi_{ 851 }(484, a) 1111e(1318)e\left(\frac{13}{18}\right)e(79)e\left(\frac{7}{9}\right)e(49)e\left(\frac{4}{9}\right)e(1118)e\left(\frac{11}{18}\right)1-1e(19)e\left(\frac{1}{9}\right)e(16)e\left(\frac{1}{6}\right)e(59)e\left(\frac{5}{9}\right)e(13)e\left(\frac{1}{3}\right)e(23)e\left(\frac{2}{3}\right)
sage: chi.jacobi_sum(n)
 
χ851(484,a)   \chi_{ 851 }(484,a) \; at   a=\;a = e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
τa(χ851(484,))   \tau_{ a }( \chi_{ 851 }(484,·) )\; at   a=\;a = e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
J(χ851(484,),χ851(n,))   J(\chi_{ 851 }(484,·),\chi_{ 851 }(n,·)) \; for   n= \; n = e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
K(a,b,χ851(484,))  K(a,b,\chi_{ 851 }(484,·)) \; at   a,b=\; a,b = e.g. 1,2