Properties

Label 3381.395
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([231,11,84]))
 
pari: [g,chi] = znchar(Mod(395,3381))
 

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.cj

\(\chi_{3381}(26,\cdot)\) \(\chi_{3381}(59,\cdot)\) \(\chi_{3381}(101,\cdot)\) \(\chi_{3381}(110,\cdot)\) \(\chi_{3381}(131,\cdot)\) \(\chi_{3381}(164,\cdot)\) \(\chi_{3381}(173,\cdot)\) \(\chi_{3381}(236,\cdot)\) \(\chi_{3381}(248,\cdot)\) \(\chi_{3381}(257,\cdot)\) \(\chi_{3381}(269,\cdot)\) \(\chi_{3381}(278,\cdot)\) \(\chi_{3381}(311,\cdot)\) \(\chi_{3381}(353,\cdot)\) \(\chi_{3381}(395,\cdot)\) \(\chi_{3381}(404,\cdot)\) \(\chi_{3381}(416,\cdot)\) \(\chi_{3381}(446,\cdot)\) \(\chi_{3381}(542,\cdot)\) \(\chi_{3381}(584,\cdot)\) \(\chi_{3381}(593,\cdot)\) \(\chi_{3381}(614,\cdot)\) \(\chi_{3381}(647,\cdot)\) \(\chi_{3381}(698,\cdot)\) \(\chi_{3381}(719,\cdot)\) \(\chi_{3381}(731,\cdot)\) \(\chi_{3381}(740,\cdot)\) \(\chi_{3381}(752,\cdot)\) \(\chi_{3381}(761,\cdot)\) \(\chi_{3381}(794,\cdot)\) ...

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

\((2255,346,442)\) → \((-1,e\left(\frac{1}{42}\right),e\left(\frac{2}{11}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\(1\)\(1\)\(e\left(\frac{223}{462}\right)\)\(e\left(\frac{223}{231}\right)\)\(e\left(\frac{86}{231}\right)\)\(e\left(\frac{69}{154}\right)\)\(e\left(\frac{395}{462}\right)\)\(e\left(\frac{41}{462}\right)\)\(e\left(\frac{51}{154}\right)\)\(e\left(\frac{215}{231}\right)\)\(e\left(\frac{85}{231}\right)\)\(e\left(\frac{37}{66}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3381 }(395,a) \;\) at \(\;a = \) e.g. 2