Properties

Label 3724.165
Modulus $3724$
Conductor $133$
Order $18$
Real no
Primitive no
Minimal no
Parity odd

Related objects

Learn more

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(3724, base_ring=CyclotomicField(18))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,12,5]))
 
pari: [g,chi] = znchar(Mod(165,3724))
 

Basic properties

Modulus: \(3724\)
Conductor: \(133\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(18\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{133}(32,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3724.ct

\(\chi_{3724}(165,\cdot)\) \(\chi_{3724}(1929,\cdot)\) \(\chi_{3724}(1941,\cdot)\) \(\chi_{3724}(2529,\cdot)\) \(\chi_{3724}(2713,\cdot)\) \(\chi_{3724}(2921,\cdot)\)

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: \(\Q(\zeta_{9})\)
Fixed field: 18.0.75855608471185389708554302866739.1

Values on generators

\((1863,3041,3137)\) → \((1,e\left(\frac{2}{3}\right),e\left(\frac{5}{18}\right))\)

Values

\(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(23\)\(25\)\(27\)
\(-1\)\(1\)\(e\left(\frac{5}{18}\right)\)\(e\left(\frac{7}{9}\right)\)\(e\left(\frac{5}{9}\right)\)\(1\)\(e\left(\frac{7}{18}\right)\)\(e\left(\frac{1}{18}\right)\)\(e\left(\frac{4}{9}\right)\)\(e\left(\frac{8}{9}\right)\)\(e\left(\frac{5}{9}\right)\)\(e\left(\frac{5}{6}\right)\)
value at e.g. 2