Properties

Label 507.482
Modulus $507$
Conductor $507$
Order $26$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(507, base_ring=CyclotomicField(26))
 
M = H._module
 
chi = DirichletCharacter(H, M([13,16]))
 
pari: [g,chi] = znchar(Mod(482,507))
 

Basic properties

Modulus: \(507\)
Conductor: \(507\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(26\)
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 507.o

\(\chi_{507}(14,\cdot)\) \(\chi_{507}(53,\cdot)\) \(\chi_{507}(92,\cdot)\) \(\chi_{507}(131,\cdot)\) \(\chi_{507}(209,\cdot)\) \(\chi_{507}(248,\cdot)\) \(\chi_{507}(287,\cdot)\) \(\chi_{507}(326,\cdot)\) \(\chi_{507}(365,\cdot)\) \(\chi_{507}(404,\cdot)\) \(\chi_{507}(443,\cdot)\) \(\chi_{507}(482,\cdot)\)

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: \(\Q(\zeta_{13})\)
Fixed field: 26.0.469739652406953148168948870108145354763666849944084892337683.1

Values on generators

\((170,340)\) → \((-1,e\left(\frac{8}{13}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(14\)\(16\)\(17\)
\( \chi_{ 507 }(482, a) \) \(-1\)\(1\)\(e\left(\frac{3}{26}\right)\)\(e\left(\frac{3}{13}\right)\)\(e\left(\frac{1}{26}\right)\)\(e\left(\frac{11}{13}\right)\)\(e\left(\frac{9}{26}\right)\)\(e\left(\frac{2}{13}\right)\)\(e\left(\frac{23}{26}\right)\)\(e\left(\frac{25}{26}\right)\)\(e\left(\frac{6}{13}\right)\)\(e\left(\frac{9}{26}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 507 }(482,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 507 }(482,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 507 }(482,·),\chi_{ 507 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 507 }(482,·)) \;\) at \(\; a,b = \) e.g. 1,2