Properties

Label 373.158
Modulus $373$
Conductor $373$
Order $62$
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([13]))
 
pari: [g,chi] = znchar(Mod(158,373))
 

Basic properties

Modulus: \(373\)
Conductor: \(373\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(62\)
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.h

\(\chi_{373}(7,\cdot)\) \(\chi_{373}(13,\cdot)\) \(\chi_{373}(17,\cdot)\) \(\chi_{373}(22,\cdot)\) \(\chi_{373}(27,\cdot)\) \(\chi_{373}(31,\cdot)\) \(\chi_{373}(55,\cdot)\) \(\chi_{373}(64,\cdot)\) \(\chi_{373}(84,\cdot)\) \(\chi_{373}(87,\cdot)\) \(\chi_{373}(137,\cdot)\) \(\chi_{373}(152,\cdot)\) \(\chi_{373}(156,\cdot)\) \(\chi_{373}(158,\cdot)\) \(\chi_{373}(160,\cdot)\) \(\chi_{373}(184,\cdot)\) \(\chi_{373}(204,\cdot)\) \(\chi_{373}(210,\cdot)\) \(\chi_{373}(219,\cdot)\) \(\chi_{373}(229,\cdot)\) \(\chi_{373}(254,\cdot)\) \(\chi_{373}(262,\cdot)\) \(\chi_{373}(264,\cdot)\) \(\chi_{373}(282,\cdot)\) \(\chi_{373}(287,\cdot)\) \(\chi_{373}(298,\cdot)\) \(\chi_{373}(324,\cdot)\) \(\chi_{373}(332,\cdot)\) \(\chi_{373}(343,\cdot)\) \(\chi_{373}(361,\cdot)\)

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 62 polynomial

Values on generators

\(2\) → \(e\left(\frac{13}{62}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 373 }(158, a) \) \(1\)\(1\)\(e\left(\frac{13}{62}\right)\)\(e\left(\frac{28}{31}\right)\)\(e\left(\frac{13}{31}\right)\)\(e\left(\frac{27}{62}\right)\)\(e\left(\frac{7}{62}\right)\)\(e\left(\frac{16}{31}\right)\)\(e\left(\frac{39}{62}\right)\)\(e\left(\frac{25}{31}\right)\)\(e\left(\frac{20}{31}\right)\)\(e\left(\frac{23}{62}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 373 }(158,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 373 }(158,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 373 }(158,·),\chi_{ 373 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 373 }(158,·)) \;\) at \(\; a,b = \) e.g. 1,2