Properties

Label 2736.2663
Modulus $2736$
Conductor $456$
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(2736, base_ring=CyclotomicField(18))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([9,9,9,13]))
 
pari: [g,chi] = znchar(Mod(2663,2736))
 

Basic properties

Modulus: \(2736\)
Conductor: \(456\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(18\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{456}(155,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2736.fy

\(\chi_{2736}(71,\cdot)\) \(\chi_{2736}(1079,\cdot)\) \(\chi_{2736}(1511,\cdot)\) \(\chi_{2736}(1655,\cdot)\) \(\chi_{2736}(1799,\cdot)\) \(\chi_{2736}(2663,\cdot)\)

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: 18.0.14478127324240404768365927869710336.1

Values on generators

\((1711,2053,1217,1009)\) → \((-1,-1,-1,e\left(\frac{13}{18}\right))\)

Values

\(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(23\)\(25\)\(29\)\(31\)\(35\)
\(-1\)\(1\)\(e\left(\frac{5}{9}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{1}{9}\right)\)\(e\left(\frac{13}{18}\right)\)\(e\left(\frac{4}{9}\right)\)\(e\left(\frac{1}{9}\right)\)\(e\left(\frac{5}{18}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{7}{18}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2736 }(2663,a) \;\) at \(\;a = \) e.g. 2