Properties

Label 389.22
Modulus $389$
Conductor $389$
Order $388$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(389, base_ring=CyclotomicField(388))
 
M = H._module
 
chi = DirichletCharacter(H, M([173]))
 
pari: [g,chi] = znchar(Mod(22,389))
 

Basic properties

Modulus: \(389\)
Conductor: \(389\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(388\)
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 389.f

\(\chi_{389}(2,\cdot)\) \(\chi_{389}(3,\cdot)\) \(\chi_{389}(8,\cdot)\) \(\chi_{389}(10,\cdot)\) \(\chi_{389}(12,\cdot)\) \(\chi_{389}(14,\cdot)\) \(\chi_{389}(15,\cdot)\) \(\chi_{389}(18,\cdot)\) \(\chi_{389}(21,\cdot)\) \(\chi_{389}(22,\cdot)\) \(\chi_{389}(23,\cdot)\) \(\chi_{389}(26,\cdot)\) \(\chi_{389}(27,\cdot)\) \(\chi_{389}(29,\cdot)\) \(\chi_{389}(31,\cdot)\) \(\chi_{389}(32,\cdot)\) \(\chi_{389}(33,\cdot)\) \(\chi_{389}(34,\cdot)\) \(\chi_{389}(37,\cdot)\) \(\chi_{389}(38,\cdot)\) \(\chi_{389}(39,\cdot)\) \(\chi_{389}(40,\cdot)\) \(\chi_{389}(43,\cdot)\) \(\chi_{389}(47,\cdot)\) \(\chi_{389}(48,\cdot)\) \(\chi_{389}(50,\cdot)\) \(\chi_{389}(51,\cdot)\) \(\chi_{389}(53,\cdot)\) \(\chi_{389}(56,\cdot)\) \(\chi_{389}(57,\cdot)\) ...

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: $\Q(\zeta_{388})$
Fixed field: Number field defined by a degree 388 polynomial (not computed)

Values on generators

\(2\) → \(e\left(\frac{173}{388}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 389 }(22, a) \) \(-1\)\(1\)\(e\left(\frac{173}{388}\right)\)\(e\left(\frac{323}{388}\right)\)\(e\left(\frac{173}{194}\right)\)\(e\left(\frac{44}{97}\right)\)\(e\left(\frac{27}{97}\right)\)\(e\left(\frac{24}{97}\right)\)\(e\left(\frac{131}{388}\right)\)\(e\left(\frac{129}{194}\right)\)\(e\left(\frac{349}{388}\right)\)\(e\left(\frac{67}{97}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 389 }(22,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 389 }(22,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 389 }(22,·),\chi_{ 389 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 389 }(22,·)) \;\) at \(\; a,b = \) e.g. 1,2