Properties

Label 161.156
Modulus $161$
Conductor $161$
Order $33$
Real no
Primitive yes
Minimal yes
Parity even

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([22,36]))
 
pari: [g,chi] = znchar(Mod(156,161))
 

Basic properties

Modulus: \(161\)
Conductor: \(161\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(33\)
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 161.m

\(\chi_{161}(2,\cdot)\) \(\chi_{161}(4,\cdot)\) \(\chi_{161}(9,\cdot)\) \(\chi_{161}(16,\cdot)\) \(\chi_{161}(18,\cdot)\) \(\chi_{161}(25,\cdot)\) \(\chi_{161}(32,\cdot)\) \(\chi_{161}(39,\cdot)\) \(\chi_{161}(58,\cdot)\) \(\chi_{161}(72,\cdot)\) \(\chi_{161}(81,\cdot)\) \(\chi_{161}(95,\cdot)\) \(\chi_{161}(100,\cdot)\) \(\chi_{161}(121,\cdot)\) \(\chi_{161}(123,\cdot)\) \(\chi_{161}(128,\cdot)\) \(\chi_{161}(142,\cdot)\) \(\chi_{161}(144,\cdot)\) \(\chi_{161}(151,\cdot)\) \(\chi_{161}(156,\cdot)\)

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: 33.33.277966181338944111003326058293667039541136678070715028736001.1

Values on generators

\((24,120)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{6}{11}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\(1\)\(1\)\(e\left(\frac{25}{33}\right)\)\(e\left(\frac{2}{33}\right)\)\(e\left(\frac{17}{33}\right)\)\(e\left(\frac{7}{33}\right)\)\(e\left(\frac{9}{11}\right)\)\(e\left(\frac{3}{11}\right)\)\(e\left(\frac{4}{33}\right)\)\(e\left(\frac{32}{33}\right)\)\(e\left(\frac{8}{33}\right)\)\(e\left(\frac{19}{33}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 161 }(156,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 161 }(156,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 161 }(156,·),\chi_{ 161 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 161 }(156,·)) \;\) at \(\; a,b = \) e.g. 1,2