Properties

Label 361.11
Modulus $361$
Conductor $361$
Order $57$
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(114))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([34]))
 
pari: [g,chi] = znchar(Mod(11,361))
 

Basic properties

Modulus: \(361\)
Conductor: \(361\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(57\)
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.i

\(\chi_{361}(7,\cdot)\) \(\chi_{361}(11,\cdot)\) \(\chi_{361}(26,\cdot)\) \(\chi_{361}(30,\cdot)\) \(\chi_{361}(45,\cdot)\) \(\chi_{361}(49,\cdot)\) \(\chi_{361}(64,\cdot)\) \(\chi_{361}(83,\cdot)\) \(\chi_{361}(87,\cdot)\) \(\chi_{361}(102,\cdot)\) \(\chi_{361}(106,\cdot)\) \(\chi_{361}(121,\cdot)\) \(\chi_{361}(125,\cdot)\) \(\chi_{361}(140,\cdot)\) \(\chi_{361}(144,\cdot)\) \(\chi_{361}(159,\cdot)\) \(\chi_{361}(163,\cdot)\) \(\chi_{361}(178,\cdot)\) \(\chi_{361}(182,\cdot)\) \(\chi_{361}(197,\cdot)\) \(\chi_{361}(201,\cdot)\) \(\chi_{361}(216,\cdot)\) \(\chi_{361}(220,\cdot)\) \(\chi_{361}(235,\cdot)\) \(\chi_{361}(239,\cdot)\) \(\chi_{361}(254,\cdot)\) \(\chi_{361}(258,\cdot)\) \(\chi_{361}(273,\cdot)\) \(\chi_{361}(277,\cdot)\) \(\chi_{361}(296,\cdot)\) ...

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: $\Q(\zeta_{57})$
Fixed field: Number field defined by a degree 57 polynomial

Values on generators

\(2\) → \(e\left(\frac{17}{57}\right)\)

Values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 361 }(11, a) \) \(1\)\(1\)\(e\left(\frac{17}{57}\right)\)\(e\left(\frac{26}{57}\right)\)\(e\left(\frac{34}{57}\right)\)\(e\left(\frac{11}{57}\right)\)\(e\left(\frac{43}{57}\right)\)\(e\left(\frac{14}{19}\right)\)\(e\left(\frac{17}{19}\right)\)\(e\left(\frac{52}{57}\right)\)\(e\left(\frac{28}{57}\right)\)\(e\left(\frac{8}{19}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 361 }(11,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 361 }(11,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 361 }(11,·),\chi_{ 361 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 361 }(11,·)) \;\) at \(\; a,b = \) e.g. 1,2