Properties

Label 361.6
Modulus $361$
Conductor $361$
Order $171$
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(342))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([140]))
 
pari: [g,chi] = znchar(Mod(6,361))
 

Basic properties

Modulus: \(361\)
Conductor: \(361\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(171\)
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.k

\(\chi_{361}(4,\cdot)\) \(\chi_{361}(5,\cdot)\) \(\chi_{361}(6,\cdot)\) \(\chi_{361}(9,\cdot)\) \(\chi_{361}(16,\cdot)\) \(\chi_{361}(17,\cdot)\) \(\chi_{361}(23,\cdot)\) \(\chi_{361}(24,\cdot)\) \(\chi_{361}(25,\cdot)\) \(\chi_{361}(35,\cdot)\) \(\chi_{361}(36,\cdot)\) \(\chi_{361}(42,\cdot)\) \(\chi_{361}(43,\cdot)\) \(\chi_{361}(44,\cdot)\) \(\chi_{361}(47,\cdot)\) \(\chi_{361}(55,\cdot)\) \(\chi_{361}(61,\cdot)\) \(\chi_{361}(63,\cdot)\) \(\chi_{361}(66,\cdot)\) \(\chi_{361}(73,\cdot)\) \(\chi_{361}(74,\cdot)\) \(\chi_{361}(80,\cdot)\) \(\chi_{361}(81,\cdot)\) \(\chi_{361}(82,\cdot)\) \(\chi_{361}(85,\cdot)\) \(\chi_{361}(92,\cdot)\) \(\chi_{361}(93,\cdot)\) \(\chi_{361}(100,\cdot)\) \(\chi_{361}(101,\cdot)\) \(\chi_{361}(104,\cdot)\) ...

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: $\Q(\zeta_{171})$
Fixed field: Number field defined by a degree 171 polynomial (not computed)

Values on generators

\(2\) → \(e\left(\frac{70}{171}\right)\)

Values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 361 }(6, a) \) \(1\)\(1\)\(e\left(\frac{70}{171}\right)\)\(e\left(\frac{154}{171}\right)\)\(e\left(\frac{140}{171}\right)\)\(e\left(\frac{166}{171}\right)\)\(e\left(\frac{53}{171}\right)\)\(e\left(\frac{23}{57}\right)\)\(e\left(\frac{13}{57}\right)\)\(e\left(\frac{137}{171}\right)\)\(e\left(\frac{65}{171}\right)\)\(e\left(\frac{43}{57}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 361 }(6,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 361 }(6,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 361 }(6,·),\chi_{ 361 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 361 }(6,·)) \;\) at \(\; a,b = \) e.g. 1,2