Properties

Label 3381.2
Modulus $3381$
Conductor $3381$
Order $462$
Real no
Primitive yes
Minimal yes
Parity odd

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([231,286,42]))
 
pari: [g,chi] = znchar(Mod(2,3381))
 

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: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3381.cl

\(\chi_{3381}(2,\cdot)\) \(\chi_{3381}(32,\cdot)\) \(\chi_{3381}(95,\cdot)\) \(\chi_{3381}(170,\cdot)\) \(\chi_{3381}(179,\cdot)\) \(\chi_{3381}(200,\cdot)\) \(\chi_{3381}(233,\cdot)\) \(\chi_{3381}(242,\cdot)\) \(\chi_{3381}(284,\cdot)\) \(\chi_{3381}(305,\cdot)\) \(\chi_{3381}(317,\cdot)\) \(\chi_{3381}(326,\cdot)\) \(\chi_{3381}(338,\cdot)\) \(\chi_{3381}(347,\cdot)\) \(\chi_{3381}(380,\cdot)\) \(\chi_{3381}(443,\cdot)\) \(\chi_{3381}(464,\cdot)\) \(\chi_{3381}(473,\cdot)\) \(\chi_{3381}(485,\cdot)\) \(\chi_{3381}(515,\cdot)\) \(\chi_{3381}(578,\cdot)\) \(\chi_{3381}(611,\cdot)\) \(\chi_{3381}(653,\cdot)\) \(\chi_{3381}(662,\cdot)\) \(\chi_{3381}(683,\cdot)\) \(\chi_{3381}(725,\cdot)\) \(\chi_{3381}(767,\cdot)\) \(\chi_{3381}(788,\cdot)\) \(\chi_{3381}(800,\cdot)\) \(\chi_{3381}(809,\cdot)\) ...

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

\((2255,346,442)\) → \((-1,e\left(\frac{13}{21}\right),e\left(\frac{1}{11}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\(-1\)\(1\)\(e\left(\frac{359}{462}\right)\)\(e\left(\frac{128}{231}\right)\)\(e\left(\frac{251}{462}\right)\)\(e\left(\frac{51}{154}\right)\)\(e\left(\frac{74}{231}\right)\)\(e\left(\frac{37}{462}\right)\)\(e\left(\frac{54}{77}\right)\)\(e\left(\frac{25}{231}\right)\)\(e\left(\frac{283}{462}\right)\)\(e\left(\frac{1}{33}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3381 }(2,a) \;\) at \(\;a = \) e.g. 2