Properties

Label 373.49
Modulus $373$
Conductor $373$
Order $31$
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(62))
 
M = H._module
 
chi = DirichletCharacter(H, M([4]))
 
pari: [g,chi] = znchar(Mod(49,373))
 

Basic properties

Modulus: \(373\)
Conductor: \(373\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(31\)
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.g

\(\chi_{373}(12,\cdot)\) \(\chi_{373}(30,\cdot)\) \(\chi_{373}(41,\cdot)\) \(\chi_{373}(49,\cdot)\) \(\chi_{373}(75,\cdot)\) \(\chi_{373}(86,\cdot)\) \(\chi_{373}(91,\cdot)\) \(\chi_{373}(109,\cdot)\) \(\chi_{373}(111,\cdot)\) \(\chi_{373}(119,\cdot)\) \(\chi_{373}(144,\cdot)\) \(\chi_{373}(154,\cdot)\) \(\chi_{373}(163,\cdot)\) \(\chi_{373}(169,\cdot)\) \(\chi_{373}(189,\cdot)\) \(\chi_{373}(213,\cdot)\) \(\chi_{373}(215,\cdot)\) \(\chi_{373}(217,\cdot)\) \(\chi_{373}(221,\cdot)\) \(\chi_{373}(236,\cdot)\) \(\chi_{373}(286,\cdot)\) \(\chi_{373}(289,\cdot)\) \(\chi_{373}(309,\cdot)\) \(\chi_{373}(318,\cdot)\) \(\chi_{373}(342,\cdot)\) \(\chi_{373}(346,\cdot)\) \(\chi_{373}(351,\cdot)\) \(\chi_{373}(356,\cdot)\) \(\chi_{373}(360,\cdot)\) \(\chi_{373}(366,\cdot)\)

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: $\Q(\zeta_{31})$
Fixed field: Number field defined by a degree 31 polynomial

Values on generators

\(2\) → \(e\left(\frac{2}{31}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 373 }(49, a) \) \(1\)\(1\)\(e\left(\frac{2}{31}\right)\)\(e\left(\frac{11}{31}\right)\)\(e\left(\frac{4}{31}\right)\)\(e\left(\frac{28}{31}\right)\)\(e\left(\frac{13}{31}\right)\)\(e\left(\frac{24}{31}\right)\)\(e\left(\frac{6}{31}\right)\)\(e\left(\frac{22}{31}\right)\)\(e\left(\frac{30}{31}\right)\)\(e\left(\frac{25}{31}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 373 }(49,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 373 }(49,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 373 }(49,·),\chi_{ 373 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 373 }(49,·)) \;\) at \(\; a,b = \) e.g. 1,2