Properties

Label 3381.53
Modulus $3381$
Conductor $3381$
Order $462$
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(3381, base_ring=CyclotomicField(462))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([231,110,399]))
 
pari: [g,chi] = znchar(Mod(53,3381))
 

Basic properties

Modulus: \(3381\)
Conductor: \(3381\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(462\)
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 3381.ci

\(\chi_{3381}(11,\cdot)\) \(\chi_{3381}(44,\cdot)\) \(\chi_{3381}(53,\cdot)\) \(\chi_{3381}(65,\cdot)\) \(\chi_{3381}(74,\cdot)\) \(\chi_{3381}(86,\cdot)\) \(\chi_{3381}(107,\cdot)\) \(\chi_{3381}(149,\cdot)\) \(\chi_{3381}(158,\cdot)\) \(\chi_{3381}(191,\cdot)\) \(\chi_{3381}(212,\cdot)\) \(\chi_{3381}(221,\cdot)\) \(\chi_{3381}(296,\cdot)\) \(\chi_{3381}(359,\cdot)\) \(\chi_{3381}(389,\cdot)\) \(\chi_{3381}(401,\cdot)\) \(\chi_{3381}(431,\cdot)\) \(\chi_{3381}(452,\cdot)\) \(\chi_{3381}(494,\cdot)\) \(\chi_{3381}(527,\cdot)\) \(\chi_{3381}(536,\cdot)\) \(\chi_{3381}(548,\cdot)\) \(\chi_{3381}(590,\cdot)\) \(\chi_{3381}(632,\cdot)\) \(\chi_{3381}(641,\cdot)\) \(\chi_{3381}(674,\cdot)\) \(\chi_{3381}(695,\cdot)\) \(\chi_{3381}(746,\cdot)\) \(\chi_{3381}(779,\cdot)\) \(\chi_{3381}(842,\cdot)\) ...

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: $\Q(\zeta_{231})$
Fixed field: Number field defined by a degree 462 polynomial (not computed)

Values on generators

\((2255,346,442)\) → \((-1,e\left(\frac{5}{21}\right),e\left(\frac{19}{22}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\(1\)\(1\)\(e\left(\frac{193}{462}\right)\)\(e\left(\frac{193}{231}\right)\)\(e\left(\frac{62}{231}\right)\)\(e\left(\frac{39}{154}\right)\)\(e\left(\frac{317}{462}\right)\)\(e\left(\frac{184}{231}\right)\)\(e\left(\frac{73}{77}\right)\)\(e\left(\frac{155}{231}\right)\)\(e\left(\frac{115}{231}\right)\)\(e\left(\frac{19}{66}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3381 }(53,a) \;\) at \(\;a = \) e.g. 2