Properties

Label 67.24
Modulus $67$
Conductor $67$
Order $11$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(67, base_ring=CyclotomicField(22))
 
M = H._module
 
chi = DirichletCharacter(H, M([14]))
 
pari: [g,chi] = znchar(Mod(24,67))
 

Basic properties

Modulus: \(67\)
Conductor: \(67\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(11\)
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 67.e

\(\chi_{67}(9,\cdot)\) \(\chi_{67}(14,\cdot)\) \(\chi_{67}(15,\cdot)\) \(\chi_{67}(22,\cdot)\) \(\chi_{67}(24,\cdot)\) \(\chi_{67}(25,\cdot)\) \(\chi_{67}(40,\cdot)\) \(\chi_{67}(59,\cdot)\) \(\chi_{67}(62,\cdot)\) \(\chi_{67}(64,\cdot)\)

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: 11.11.1822837804551761449.1

Values on generators

\(2\) → \(e\left(\frac{7}{11}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 67 }(24, a) \) \(1\)\(1\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{9}{11}\right)\)\(e\left(\frac{3}{11}\right)\)\(e\left(\frac{6}{11}\right)\)\(e\left(\frac{5}{11}\right)\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{10}{11}\right)\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{2}{11}\right)\)\(e\left(\frac{6}{11}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 67 }(24,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 67 }(24,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 67 }(24,·),\chi_{ 67 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 67 }(24,·)) \;\) at \(\; a,b = \) e.g. 1,2