Properties

Label 3381.37
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([0,352,441]))
 
pari: [g,chi] = znchar(Mod(37,3381))
 

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 \(\chi_{1127}(37,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3381.ch

\(\chi_{3381}(37,\cdot)\) \(\chi_{3381}(88,\cdot)\) \(\chi_{3381}(109,\cdot)\) \(\chi_{3381}(130,\cdot)\) \(\chi_{3381}(172,\cdot)\) \(\chi_{3381}(205,\cdot)\) \(\chi_{3381}(235,\cdot)\) \(\chi_{3381}(247,\cdot)\) \(\chi_{3381}(268,\cdot)\) \(\chi_{3381}(310,\cdot)\) \(\chi_{3381}(319,\cdot)\) \(\chi_{3381}(352,\cdot)\) \(\chi_{3381}(382,\cdot)\) \(\chi_{3381}(424,\cdot)\) \(\chi_{3381}(457,\cdot)\) \(\chi_{3381}(550,\cdot)\) \(\chi_{3381}(562,\cdot)\) \(\chi_{3381}(571,\cdot)\) \(\chi_{3381}(592,\cdot)\) \(\chi_{3381}(613,\cdot)\) \(\chi_{3381}(688,\cdot)\) \(\chi_{3381}(697,\cdot)\) \(\chi_{3381}(709,\cdot)\) \(\chi_{3381}(718,\cdot)\) \(\chi_{3381}(730,\cdot)\) \(\chi_{3381}(751,\cdot)\) \(\chi_{3381}(793,\cdot)\) \(\chi_{3381}(835,\cdot)\) \(\chi_{3381}(856,\cdot)\) \(\chi_{3381}(865,\cdot)\) ...

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

\((2255,346,442)\) → \((1,e\left(\frac{16}{21}\right),e\left(\frac{21}{22}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\(-1\)\(1\)\(e\left(\frac{166}{231}\right)\)\(e\left(\frac{101}{231}\right)\)\(e\left(\frac{23}{462}\right)\)\(e\left(\frac{12}{77}\right)\)\(e\left(\frac{355}{462}\right)\)\(e\left(\frac{31}{462}\right)\)\(e\left(\frac{39}{77}\right)\)\(e\left(\frac{202}{231}\right)\)\(e\left(\frac{337}{462}\right)\)\(e\left(\frac{65}{66}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3381 }(37,a) \;\) at \(\;a = \) e.g. 2