Properties

Label 361.58
Modulus $361$
Conductor $361$
Order $19$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(361, base_ring=CyclotomicField(38))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([2]))
 
pari: [g,chi] = znchar(Mod(58,361))
 

Basic properties

Modulus: \(361\)
Conductor: \(361\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(19\)
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 361.g

\(\chi_{361}(20,\cdot)\) \(\chi_{361}(39,\cdot)\) \(\chi_{361}(58,\cdot)\) \(\chi_{361}(77,\cdot)\) \(\chi_{361}(96,\cdot)\) \(\chi_{361}(115,\cdot)\) \(\chi_{361}(134,\cdot)\) \(\chi_{361}(153,\cdot)\) \(\chi_{361}(172,\cdot)\) \(\chi_{361}(191,\cdot)\) \(\chi_{361}(210,\cdot)\) \(\chi_{361}(229,\cdot)\) \(\chi_{361}(248,\cdot)\) \(\chi_{361}(267,\cdot)\) \(\chi_{361}(286,\cdot)\) \(\chi_{361}(305,\cdot)\) \(\chi_{361}(324,\cdot)\) \(\chi_{361}(343,\cdot)\)

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: \(\Q(\zeta_{19})\)
Fixed field: 19.19.10842505080063916320800450434338728415281531281.1

Values on generators

\(2\) → \(e\left(\frac{1}{19}\right)\)

Values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 361 }(58, a) \) \(1\)\(1\)\(e\left(\frac{1}{19}\right)\)\(e\left(\frac{6}{19}\right)\)\(e\left(\frac{2}{19}\right)\)\(e\left(\frac{4}{19}\right)\)\(e\left(\frac{7}{19}\right)\)\(e\left(\frac{17}{19}\right)\)\(e\left(\frac{3}{19}\right)\)\(e\left(\frac{12}{19}\right)\)\(e\left(\frac{5}{19}\right)\)\(e\left(\frac{7}{19}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 361 }(58,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 361 }(58,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 361 }(58,·),\chi_{ 361 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 361 }(58,·)) \;\) at \(\; a,b = \) e.g. 1,2