Properties

Label 89.68
Modulus $89$
Conductor $89$
Order $44$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(89, base_ring=CyclotomicField(44))
 
M = H._module
 
chi = DirichletCharacter(H, M([19]))
 
pari: [g,chi] = znchar(Mod(68,89))
 

Basic properties

Modulus: \(89\)
Conductor: \(89\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(44\)
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 89.g

\(\chi_{89}(5,\cdot)\) \(\chi_{89}(9,\cdot)\) \(\chi_{89}(10,\cdot)\) \(\chi_{89}(17,\cdot)\) \(\chi_{89}(18,\cdot)\) \(\chi_{89}(20,\cdot)\) \(\chi_{89}(21,\cdot)\) \(\chi_{89}(36,\cdot)\) \(\chi_{89}(40,\cdot)\) \(\chi_{89}(42,\cdot)\) \(\chi_{89}(47,\cdot)\) \(\chi_{89}(49,\cdot)\) \(\chi_{89}(53,\cdot)\) \(\chi_{89}(68,\cdot)\) \(\chi_{89}(69,\cdot)\) \(\chi_{89}(71,\cdot)\) \(\chi_{89}(72,\cdot)\) \(\chi_{89}(79,\cdot)\) \(\chi_{89}(80,\cdot)\) \(\chi_{89}(84,\cdot)\)

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: \(\Q(\zeta_{44})\)
Fixed field: Number field defined by a degree 44 polynomial

Values on generators

\(3\) → \(e\left(\frac{19}{44}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 89 }(68, a) \) \(1\)\(1\)\(e\left(\frac{10}{11}\right)\)\(e\left(\frac{19}{44}\right)\)\(e\left(\frac{9}{11}\right)\)\(e\left(\frac{5}{22}\right)\)\(e\left(\frac{15}{44}\right)\)\(e\left(\frac{43}{44}\right)\)\(e\left(\frac{8}{11}\right)\)\(e\left(\frac{19}{22}\right)\)\(e\left(\frac{3}{22}\right)\)\(e\left(\frac{3}{11}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 89 }(68,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 89 }(68,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 89 }(68,·),\chi_{ 89 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 89 }(68,·)) \;\) at \(\; a,b = \) e.g. 1,2