Properties

Label 69.65
Modulus $69$
Conductor $69$
Order $22$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(69, base_ring=CyclotomicField(22))
 
M = H._module
 
chi = DirichletCharacter(H, M([11,15]))
 
pari: [g,chi] = znchar(Mod(65,69))
 

Basic properties

Modulus: \(69\)
Conductor: \(69\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(22\)
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 69.g

\(\chi_{69}(5,\cdot)\) \(\chi_{69}(11,\cdot)\) \(\chi_{69}(14,\cdot)\) \(\chi_{69}(17,\cdot)\) \(\chi_{69}(20,\cdot)\) \(\chi_{69}(38,\cdot)\) \(\chi_{69}(44,\cdot)\) \(\chi_{69}(53,\cdot)\) \(\chi_{69}(56,\cdot)\) \(\chi_{69}(65,\cdot)\)

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: \(\Q(\zeta_{11})\)
Fixed field: \(\Q(\zeta_{69})^+\)

Values on generators

\((47,28)\) → \((-1,e\left(\frac{15}{22}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 69 }(65, a) \) \(1\)\(1\)\(e\left(\frac{19}{22}\right)\)\(e\left(\frac{8}{11}\right)\)\(e\left(\frac{2}{11}\right)\)\(e\left(\frac{21}{22}\right)\)\(e\left(\frac{13}{22}\right)\)\(e\left(\frac{1}{22}\right)\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{6}{11}\right)\)\(e\left(\frac{9}{11}\right)\)\(e\left(\frac{5}{11}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 69 }(65,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 69 }(65,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 69 }(65,·),\chi_{ 69 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 69 }(65,·)) \;\) at \(\; a,b = \) e.g. 1,2