Properties

Label 161.109
Modulus $161$
Conductor $161$
Order $66$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(161, base_ring=CyclotomicField(66))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([44,21]))
 
pari: [g,chi] = znchar(Mod(109,161))
 

Basic properties

Modulus: \(161\)
Conductor: \(161\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(66\)
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 161.p

\(\chi_{161}(11,\cdot)\) \(\chi_{161}(30,\cdot)\) \(\chi_{161}(37,\cdot)\) \(\chi_{161}(44,\cdot)\) \(\chi_{161}(51,\cdot)\) \(\chi_{161}(53,\cdot)\) \(\chi_{161}(60,\cdot)\) \(\chi_{161}(65,\cdot)\) \(\chi_{161}(67,\cdot)\) \(\chi_{161}(74,\cdot)\) \(\chi_{161}(79,\cdot)\) \(\chi_{161}(86,\cdot)\) \(\chi_{161}(88,\cdot)\) \(\chi_{161}(102,\cdot)\) \(\chi_{161}(107,\cdot)\) \(\chi_{161}(109,\cdot)\) \(\chi_{161}(130,\cdot)\) \(\chi_{161}(135,\cdot)\) \(\chi_{161}(149,\cdot)\) \(\chi_{161}(158,\cdot)\)

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: \(\Q(\zeta_{33})\)
Fixed field: Number field defined by a degree 66 polynomial

Values on generators

\((24,120)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{7}{22}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\(-1\)\(1\)\(e\left(\frac{32}{33}\right)\)\(e\left(\frac{25}{33}\right)\)\(e\left(\frac{31}{33}\right)\)\(e\left(\frac{43}{66}\right)\)\(e\left(\frac{8}{11}\right)\)\(e\left(\frac{10}{11}\right)\)\(e\left(\frac{17}{33}\right)\)\(e\left(\frac{41}{66}\right)\)\(e\left(\frac{35}{66}\right)\)\(e\left(\frac{23}{33}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 161 }(109,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 161 }(109,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 161 }(109,·),\chi_{ 161 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 161 }(109,·)) \;\) at \(\; a,b = \) e.g. 1,2