Properties

Label 373.24
Modulus $373$
Conductor $373$
Order $372$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(373, base_ring=CyclotomicField(372))
 
M = H._module
 
chi = DirichletCharacter(H, M([241]))
 
pari: [g,chi] = znchar(Mod(24,373))
 

Basic properties

Modulus: \(373\)
Conductor: \(373\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(372\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 373.l

\(\chi_{373}(2,\cdot)\) \(\chi_{373}(5,\cdot)\) \(\chi_{373}(6,\cdot)\) \(\chi_{373}(11,\cdot)\) \(\chi_{373}(14,\cdot)\) \(\chi_{373}(15,\cdot)\) \(\chi_{373}(24,\cdot)\) \(\chi_{373}(26,\cdot)\) \(\chi_{373}(32,\cdot)\) \(\chi_{373}(34,\cdot)\) \(\chi_{373}(35,\cdot)\) \(\chi_{373}(42,\cdot)\) \(\chi_{373}(43,\cdot)\) \(\chi_{373}(44,\cdot)\) \(\chi_{373}(47,\cdot)\) \(\chi_{373}(53,\cdot)\) \(\chi_{373}(54,\cdot)\) \(\chi_{373}(57,\cdot)\) \(\chi_{373}(60,\cdot)\) \(\chi_{373}(61,\cdot)\) \(\chi_{373}(62,\cdot)\) \(\chi_{373}(65,\cdot)\) \(\chi_{373}(72,\cdot)\) \(\chi_{373}(76,\cdot)\) \(\chi_{373}(77,\cdot)\) \(\chi_{373}(78,\cdot)\) \(\chi_{373}(79,\cdot)\) \(\chi_{373}(80,\cdot)\) \(\chi_{373}(82,\cdot)\) \(\chi_{373}(85,\cdot)\) ...

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: $\Q(\zeta_{372})$
Fixed field: Number field defined by a degree 372 polynomial (not computed)

Values on generators

\(2\) → \(e\left(\frac{241}{372}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 373 }(24, a) \) \(-1\)\(1\)\(e\left(\frac{241}{372}\right)\)\(e\left(\frac{35}{186}\right)\)\(e\left(\frac{55}{186}\right)\)\(e\left(\frac{181}{372}\right)\)\(e\left(\frac{311}{372}\right)\)\(e\left(\frac{17}{62}\right)\)\(e\left(\frac{117}{124}\right)\)\(e\left(\frac{35}{93}\right)\)\(e\left(\frac{25}{186}\right)\)\(e\left(\frac{269}{372}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 373 }(24,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 373 }(24,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 373 }(24,·),\chi_{ 373 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 373 }(24,·)) \;\) at \(\; a,b = \) e.g. 1,2