Properties

Label 3381.52
Modulus $3381$
Conductor $1127$
Order $462$
Real no
Primitive no
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([0,11,378]))
 
pari: [g,chi] = znchar(Mod(52,3381))
 

Basic properties

Modulus: \(3381\)
Conductor: \(1127\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(462\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1127}(52,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3381.cg

\(\chi_{3381}(52,\cdot)\) \(\chi_{3381}(73,\cdot)\) \(\chi_{3381}(82,\cdot)\) \(\chi_{3381}(94,\cdot)\) \(\chi_{3381}(124,\cdot)\) \(\chi_{3381}(187,\cdot)\) \(\chi_{3381}(220,\cdot)\) \(\chi_{3381}(262,\cdot)\) \(\chi_{3381}(271,\cdot)\) \(\chi_{3381}(292,\cdot)\) \(\chi_{3381}(334,\cdot)\) \(\chi_{3381}(376,\cdot)\) \(\chi_{3381}(397,\cdot)\) \(\chi_{3381}(409,\cdot)\) \(\chi_{3381}(418,\cdot)\) \(\chi_{3381}(430,\cdot)\) \(\chi_{3381}(439,\cdot)\) \(\chi_{3381}(514,\cdot)\) \(\chi_{3381}(535,\cdot)\) \(\chi_{3381}(556,\cdot)\) \(\chi_{3381}(565,\cdot)\) \(\chi_{3381}(577,\cdot)\) \(\chi_{3381}(670,\cdot)\) \(\chi_{3381}(703,\cdot)\) \(\chi_{3381}(745,\cdot)\) \(\chi_{3381}(775,\cdot)\) \(\chi_{3381}(808,\cdot)\) \(\chi_{3381}(817,\cdot)\) \(\chi_{3381}(859,\cdot)\) \(\chi_{3381}(880,\cdot)\) ...

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

\((2255,346,442)\) → \((1,e\left(\frac{1}{42}\right),e\left(\frac{9}{11}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\(-1\)\(1\)\(e\left(\frac{59}{231}\right)\)\(e\left(\frac{118}{231}\right)\)\(e\left(\frac{235}{462}\right)\)\(e\left(\frac{59}{77}\right)\)\(e\left(\frac{353}{462}\right)\)\(e\left(\frac{73}{231}\right)\)\(e\left(\frac{37}{154}\right)\)\(e\left(\frac{5}{231}\right)\)\(e\left(\frac{149}{462}\right)\)\(e\left(\frac{7}{66}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3381 }(52,a) \;\) at \(\;a = \) e.g. 2