Properties

Label 67.23
Modulus $67$
Conductor $67$
Order $33$
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(66))
 
M = H._module
 
chi = DirichletCharacter(H, M([28]))
 
pari: [g,chi] = znchar(Mod(23,67))
 

Basic properties

Modulus: \(67\)
Conductor: \(67\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(33\)
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.g

\(\chi_{67}(4,\cdot)\) \(\chi_{67}(6,\cdot)\) \(\chi_{67}(10,\cdot)\) \(\chi_{67}(16,\cdot)\) \(\chi_{67}(17,\cdot)\) \(\chi_{67}(19,\cdot)\) \(\chi_{67}(21,\cdot)\) \(\chi_{67}(23,\cdot)\) \(\chi_{67}(26,\cdot)\) \(\chi_{67}(33,\cdot)\) \(\chi_{67}(35,\cdot)\) \(\chi_{67}(36,\cdot)\) \(\chi_{67}(39,\cdot)\) \(\chi_{67}(47,\cdot)\) \(\chi_{67}(49,\cdot)\) \(\chi_{67}(54,\cdot)\) \(\chi_{67}(55,\cdot)\) \(\chi_{67}(56,\cdot)\) \(\chi_{67}(60,\cdot)\) \(\chi_{67}(65,\cdot)\)

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: \(\Q(\zeta_{33})\)
Fixed field: Number field defined by a degree 33 polynomial

Values on generators

\(2\) → \(e\left(\frac{14}{33}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 67 }(23, a) \) \(1\)\(1\)\(e\left(\frac{14}{33}\right)\)\(e\left(\frac{6}{11}\right)\)\(e\left(\frac{28}{33}\right)\)\(e\left(\frac{4}{11}\right)\)\(e\left(\frac{32}{33}\right)\)\(e\left(\frac{25}{33}\right)\)\(e\left(\frac{3}{11}\right)\)\(e\left(\frac{1}{11}\right)\)\(e\left(\frac{26}{33}\right)\)\(e\left(\frac{1}{33}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 67 }(23,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 67 }(23,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 67 }(23,·),\chi_{ 67 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 67 }(23,·)) \;\) at \(\; a,b = \) e.g. 1,2