Properties

Label 373.185
Modulus $373$
Conductor $373$
Order $124$
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(124))
 
M = H._module
 
chi = DirichletCharacter(H, M([17]))
 
pari: [g,chi] = znchar(Mod(185,373))
 

Basic properties

Modulus: \(373\)
Conductor: \(373\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(124\)
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.j

\(\chi_{373}(8,\cdot)\) \(\chi_{373}(18,\cdot)\) \(\chi_{373}(19,\cdot)\) \(\chi_{373}(20,\cdot)\) \(\chi_{373}(23,\cdot)\) \(\chi_{373}(33,\cdot)\) \(\chi_{373}(45,\cdot)\) \(\chi_{373}(50,\cdot)\) \(\chi_{373}(56,\cdot)\) \(\chi_{373}(58,\cdot)\) \(\chi_{373}(67,\cdot)\) \(\chi_{373}(74,\cdot)\) \(\chi_{373}(96,\cdot)\) \(\chi_{373}(97,\cdot)\) \(\chi_{373}(113,\cdot)\) \(\chi_{373}(125,\cdot)\) \(\chi_{373}(126,\cdot)\) \(\chi_{373}(129,\cdot)\) \(\chi_{373}(133,\cdot)\) \(\chi_{373}(136,\cdot)\) \(\chi_{373}(139,\cdot)\) \(\chi_{373}(140,\cdot)\) \(\chi_{373}(142,\cdot)\) \(\chi_{373}(145,\cdot)\) \(\chi_{373}(146,\cdot)\) \(\chi_{373}(157,\cdot)\) \(\chi_{373}(161,\cdot)\) \(\chi_{373}(167,\cdot)\) \(\chi_{373}(176,\cdot)\) \(\chi_{373}(185,\cdot)\) ...

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: $\Q(\zeta_{124})$
Fixed field: Number field defined by a degree 124 polynomial (not computed)

Values on generators

\(2\) → \(e\left(\frac{17}{124}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 373 }(185, a) \) \(-1\)\(1\)\(e\left(\frac{17}{124}\right)\)\(e\left(\frac{39}{62}\right)\)\(e\left(\frac{17}{62}\right)\)\(e\left(\frac{21}{124}\right)\)\(e\left(\frac{95}{124}\right)\)\(e\left(\frac{9}{62}\right)\)\(e\left(\frac{51}{124}\right)\)\(e\left(\frac{8}{31}\right)\)\(e\left(\frac{19}{62}\right)\)\(e\left(\frac{73}{124}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 373 }(185,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 373 }(185,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 373 }(185,·),\chi_{ 373 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 373 }(185,·)) \;\) at \(\; a,b = \) e.g. 1,2