Properties

Label 2268.397
Modulus $2268$
Conductor $189$
Order $18$
Real no
Primitive no
Minimal no
Parity odd

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(2268, base_ring=CyclotomicField(18))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,10,15]))
 
pari: [g,chi] = znchar(Mod(397,2268))
 

Basic properties

Modulus: \(2268\)
Conductor: \(189\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(18\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{189}(187,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2268.ch

\(\chi_{2268}(397,\cdot)\) \(\chi_{2268}(577,\cdot)\) \(\chi_{2268}(1153,\cdot)\) \(\chi_{2268}(1333,\cdot)\) \(\chi_{2268}(1909,\cdot)\) \(\chi_{2268}(2089,\cdot)\)

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: 18.0.4675260431318755700605016170035783.2

Values on generators

\((1135,1541,325)\) → \((1,e\left(\frac{5}{9}\right),e\left(\frac{5}{6}\right))\)

Values

\(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)
\(-1\)\(1\)\(e\left(\frac{17}{18}\right)\)\(e\left(\frac{5}{9}\right)\)\(e\left(\frac{17}{18}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{7}{9}\right)\)\(e\left(\frac{8}{9}\right)\)\(e\left(\frac{5}{9}\right)\)\(e\left(\frac{17}{18}\right)\)\(1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2268 }(397,a) \;\) at \(\;a = \) e.g. 2