Properties

Label 659.225
Modulus $659$
Conductor $659$
Order $47$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(659, base_ring=CyclotomicField(94))
 
M = H._module
 
chi = DirichletCharacter(H, M([68]))
 
pari: [g,chi] = znchar(Mod(225,659))
 

Basic properties

Modulus: \(659\)
Conductor: \(659\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(47\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 659.e

\(\chi_{659}(14,\cdot)\) \(\chi_{659}(15,\cdot)\) \(\chi_{659}(44,\cdot)\) \(\chi_{659}(57,\cdot)\) \(\chi_{659}(73,\cdot)\) \(\chi_{659}(80,\cdot)\) \(\chi_{659}(85,\cdot)\) \(\chi_{659}(108,\cdot)\) \(\chi_{659}(139,\cdot)\) \(\chi_{659}(156,\cdot)\) \(\chi_{659}(173,\cdot)\) \(\chi_{659}(185,\cdot)\) \(\chi_{659}(194,\cdot)\) \(\chi_{659}(196,\cdot)\) \(\chi_{659}(207,\cdot)\) \(\chi_{659}(210,\cdot)\) \(\chi_{659}(225,\cdot)\) \(\chi_{659}(232,\cdot)\) \(\chi_{659}(262,\cdot)\) \(\chi_{659}(274,\cdot)\) \(\chi_{659}(299,\cdot)\) \(\chi_{659}(302,\cdot)\) \(\chi_{659}(304,\cdot)\) \(\chi_{659}(323,\cdot)\) \(\chi_{659}(325,\cdot)\) \(\chi_{659}(363,\cdot)\) \(\chi_{659}(373,\cdot)\) \(\chi_{659}(436,\cdot)\) \(\chi_{659}(445,\cdot)\) \(\chi_{659}(461,\cdot)\) ...

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: $\Q(\zeta_{47})$
Fixed field: Number field defined by a degree 47 polynomial

Values on generators

\(2\) → \(e\left(\frac{34}{47}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 659 }(225, a) \) \(1\)\(1\)\(e\left(\frac{34}{47}\right)\)\(e\left(\frac{26}{47}\right)\)\(e\left(\frac{21}{47}\right)\)\(e\left(\frac{29}{47}\right)\)\(e\left(\frac{13}{47}\right)\)\(e\left(\frac{46}{47}\right)\)\(e\left(\frac{8}{47}\right)\)\(e\left(\frac{5}{47}\right)\)\(e\left(\frac{16}{47}\right)\)\(e\left(\frac{18}{47}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 659 }(225,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 659 }(225,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 659 }(225,·),\chi_{ 659 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 659 }(225,·)) \;\) at \(\; a,b = \) e.g. 1,2