Properties

Label 385.233
Modulus $385$
Conductor $385$
Order $60$
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(385, base_ring=CyclotomicField(60))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([45,20,6]))
 
pari: [g,chi] = znchar(Mod(233,385))
 

Basic properties

Modulus: \(385\)
Conductor: \(385\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(60\)
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 385.bv

\(\chi_{385}(2,\cdot)\) \(\chi_{385}(18,\cdot)\) \(\chi_{385}(72,\cdot)\) \(\chi_{385}(107,\cdot)\) \(\chi_{385}(123,\cdot)\) \(\chi_{385}(128,\cdot)\) \(\chi_{385}(172,\cdot)\) \(\chi_{385}(193,\cdot)\) \(\chi_{385}(228,\cdot)\) \(\chi_{385}(233,\cdot)\) \(\chi_{385}(277,\cdot)\) \(\chi_{385}(282,\cdot)\) \(\chi_{385}(303,\cdot)\) \(\chi_{385}(338,\cdot)\) \(\chi_{385}(347,\cdot)\) \(\chi_{385}(382,\cdot)\)

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: \(\Q(\zeta_{60})\)
Fixed field: Number field defined by a degree 60 polynomial

Values on generators

\((232,276,211)\) → \((-i,e\left(\frac{1}{3}\right),e\left(\frac{1}{10}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(12\)\(13\)\(16\)\(17\)
\(1\)\(1\)\(e\left(\frac{31}{60}\right)\)\(e\left(\frac{23}{60}\right)\)\(e\left(\frac{1}{30}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{11}{20}\right)\)\(e\left(\frac{23}{30}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{59}{60}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 385 }(233,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 385 }(233,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 385 }(233,·),\chi_{ 385 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 385 }(233,·)) \;\) at \(\; a,b = \) e.g. 1,2