Properties

Label 599.102
Modulus $599$
Conductor $599$
Order $23$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(599, base_ring=CyclotomicField(46))
 
M = H._module
 
chi = DirichletCharacter(H, M([22]))
 
pari: [g,chi] = znchar(Mod(102,599))
 

Basic properties

Modulus: \(599\)
Conductor: \(599\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(23\)
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 599.d

\(\chi_{599}(18,\cdot)\) \(\chi_{599}(39,\cdot)\) \(\chi_{599}(57,\cdot)\) \(\chi_{599}(102,\cdot)\) \(\chi_{599}(103,\cdot)\) \(\chi_{599}(151,\cdot)\) \(\chi_{599}(221,\cdot)\) \(\chi_{599}(233,\cdot)\) \(\chi_{599}(254,\cdot)\) \(\chi_{599}(322,\cdot)\) \(\chi_{599}(323,\cdot)\) \(\chi_{599}(324,\cdot)\) \(\chi_{599}(379,\cdot)\) \(\chi_{599}(384,\cdot)\) \(\chi_{599}(405,\cdot)\) \(\chi_{599}(423,\cdot)\) \(\chi_{599}(426,\cdot)\) \(\chi_{599}(427,\cdot)\) \(\chi_{599}(441,\cdot)\) \(\chi_{599}(480,\cdot)\) \(\chi_{599}(498,\cdot)\) \(\chi_{599}(578,\cdot)\)

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: \(\Q(\zeta_{23})\)
Fixed field: Number field defined by a degree 23 polynomial

Values on generators

\(7\) → \(e\left(\frac{11}{23}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 599 }(102, a) \) \(1\)\(1\)\(e\left(\frac{7}{23}\right)\)\(e\left(\frac{6}{23}\right)\)\(e\left(\frac{14}{23}\right)\)\(e\left(\frac{21}{23}\right)\)\(e\left(\frac{13}{23}\right)\)\(e\left(\frac{11}{23}\right)\)\(e\left(\frac{21}{23}\right)\)\(e\left(\frac{12}{23}\right)\)\(e\left(\frac{5}{23}\right)\)\(e\left(\frac{19}{23}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 599 }(102,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 599 }(102,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 599 }(102,·),\chi_{ 599 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 599 }(102,·)) \;\) at \(\; a,b = \) e.g. 1,2