Properties

Label 3381.19
Modulus $3381$
Conductor $161$
Order $66$
Real no
Primitive no
Minimal no
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(66))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,55,45]))
 
pari: [g,chi] = znchar(Mod(19,3381))
 

Basic properties

Modulus: \(3381\)
Conductor: \(161\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(66\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{161}(19,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3381.bv

\(\chi_{3381}(19,\cdot)\) \(\chi_{3381}(166,\cdot)\) \(\chi_{3381}(178,\cdot)\) \(\chi_{3381}(313,\cdot)\) \(\chi_{3381}(619,\cdot)\) \(\chi_{3381}(766,\cdot)\) \(\chi_{3381}(1207,\cdot)\) \(\chi_{3381}(1354,\cdot)\) \(\chi_{3381}(1489,\cdot)\) \(\chi_{3381}(1648,\cdot)\) \(\chi_{3381}(1930,\cdot)\) \(\chi_{3381}(1942,\cdot)\) \(\chi_{3381}(2077,\cdot)\) \(\chi_{3381}(2089,\cdot)\) \(\chi_{3381}(2236,\cdot)\) \(\chi_{3381}(2383,\cdot)\) \(\chi_{3381}(2518,\cdot)\) \(\chi_{3381}(2665,\cdot)\) \(\chi_{3381}(2959,\cdot)\) \(\chi_{3381}(3253,\cdot)\)

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: \(\Q(\zeta_{33})\)
Fixed field: Number field defined by a degree 66 polynomial

Values on generators

\((2255,346,442)\) → \((1,e\left(\frac{5}{6}\right),e\left(\frac{15}{22}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\(1\)\(1\)\(e\left(\frac{1}{33}\right)\)\(e\left(\frac{2}{33}\right)\)\(e\left(\frac{28}{33}\right)\)\(e\left(\frac{1}{11}\right)\)\(e\left(\frac{29}{33}\right)\)\(e\left(\frac{31}{66}\right)\)\(e\left(\frac{1}{22}\right)\)\(e\left(\frac{4}{33}\right)\)\(e\left(\frac{20}{33}\right)\)\(e\left(\frac{13}{33}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3381 }(19,a) \;\) at \(\;a = \) e.g. 2