Properties

Label 3381.4
Modulus $3381$
Conductor $1127$
Order $231$
Real no
Primitive no
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([0,110,84]))
 
pari: [g,chi] = znchar(Mod(4,3381))
 

Basic properties

Modulus: \(3381\)
Conductor: \(1127\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(231\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1127}(4,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3381.ce

\(\chi_{3381}(4,\cdot)\) \(\chi_{3381}(16,\cdot)\) \(\chi_{3381}(25,\cdot)\) \(\chi_{3381}(58,\cdot)\) \(\chi_{3381}(100,\cdot)\) \(\chi_{3381}(121,\cdot)\) \(\chi_{3381}(142,\cdot)\) \(\chi_{3381}(151,\cdot)\) \(\chi_{3381}(163,\cdot)\) \(\chi_{3381}(193,\cdot)\) \(\chi_{3381}(256,\cdot)\) \(\chi_{3381}(289,\cdot)\) \(\chi_{3381}(331,\cdot)\) \(\chi_{3381}(340,\cdot)\) \(\chi_{3381}(394,\cdot)\) \(\chi_{3381}(403,\cdot)\) \(\chi_{3381}(445,\cdot)\) \(\chi_{3381}(466,\cdot)\) \(\chi_{3381}(478,\cdot)\) \(\chi_{3381}(487,\cdot)\) \(\chi_{3381}(499,\cdot)\) \(\chi_{3381}(541,\cdot)\) \(\chi_{3381}(583,\cdot)\) \(\chi_{3381}(604,\cdot)\) \(\chi_{3381}(625,\cdot)\) \(\chi_{3381}(634,\cdot)\) \(\chi_{3381}(646,\cdot)\) \(\chi_{3381}(676,\cdot)\) \(\chi_{3381}(739,\cdot)\) \(\chi_{3381}(772,\cdot)\) ...

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 231 polynomial (not computed)

Values on generators

\((2255,346,442)\) → \((1,e\left(\frac{5}{21}\right),e\left(\frac{2}{11}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\(1\)\(1\)\(e\left(\frac{128}{231}\right)\)\(e\left(\frac{25}{231}\right)\)\(e\left(\frac{20}{231}\right)\)\(e\left(\frac{51}{77}\right)\)\(e\left(\frac{148}{231}\right)\)\(e\left(\frac{37}{231}\right)\)\(e\left(\frac{31}{77}\right)\)\(e\left(\frac{50}{231}\right)\)\(e\left(\frac{52}{231}\right)\)\(e\left(\frac{2}{33}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3381 }(4,a) \;\) at \(\;a = \) e.g. 2