Properties

Label 373.46
Modulus $373$
Conductor $373$
Order $93$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(373, base_ring=CyclotomicField(186))
 
M = H._module
 
chi = DirichletCharacter(H, M([176]))
 
pari: [g,chi] = znchar(Mod(46,373))
 

Basic properties

Modulus: \(373\)
Conductor: \(373\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(93\)
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 373.i

\(\chi_{373}(9,\cdot)\) \(\chi_{373}(16,\cdot)\) \(\chi_{373}(21,\cdot)\) \(\chi_{373}(28,\cdot)\) \(\chi_{373}(29,\cdot)\) \(\chi_{373}(38,\cdot)\) \(\chi_{373}(39,\cdot)\) \(\chi_{373}(40,\cdot)\) \(\chi_{373}(46,\cdot)\) \(\chi_{373}(51,\cdot)\) \(\chi_{373}(52,\cdot)\) \(\chi_{373}(66,\cdot)\) \(\chi_{373}(68,\cdot)\) \(\chi_{373}(70,\cdot)\) \(\chi_{373}(73,\cdot)\) \(\chi_{373}(81,\cdot)\) \(\chi_{373}(83,\cdot)\) \(\chi_{373}(93,\cdot)\) \(\chi_{373}(94,\cdot)\) \(\chi_{373}(95,\cdot)\) \(\chi_{373}(100,\cdot)\) \(\chi_{373}(101,\cdot)\) \(\chi_{373}(107,\cdot)\) \(\chi_{373}(108,\cdot)\) \(\chi_{373}(115,\cdot)\) \(\chi_{373}(124,\cdot)\) \(\chi_{373}(130,\cdot)\) \(\chi_{373}(148,\cdot)\) \(\chi_{373}(165,\cdot)\) \(\chi_{373}(170,\cdot)\) ...

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: $\Q(\zeta_{93})$
Fixed field: Number field defined by a degree 93 polynomial

Values on generators

\(2\) → \(e\left(\frac{88}{93}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 373 }(46, a) \) \(1\)\(1\)\(e\left(\frac{88}{93}\right)\)\(e\left(\frac{19}{93}\right)\)\(e\left(\frac{83}{93}\right)\)\(e\left(\frac{85}{93}\right)\)\(e\left(\frac{14}{93}\right)\)\(e\left(\frac{11}{31}\right)\)\(e\left(\frac{26}{31}\right)\)\(e\left(\frac{38}{93}\right)\)\(e\left(\frac{80}{93}\right)\)\(e\left(\frac{77}{93}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 373 }(46,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 373 }(46,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 373 }(46,·),\chi_{ 373 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 373 }(46,·)) \;\) at \(\; a,b = \) e.g. 1,2