Properties

Label 2736.553
Modulus $2736$
Conductor $1368$
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([0,9,6,1]))
 
pari: [g,chi] = znchar(Mod(553,2736))
 

Basic properties

Modulus: \(2736\)
Conductor: \(1368\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(18\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1368}(1237,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2736.fc

\(\chi_{2736}(553,\cdot)\) \(\chi_{2736}(889,\cdot)\) \(\chi_{2736}(1321,\cdot)\) \(\chi_{2736}(1561,\cdot)\) \(\chi_{2736}(1609,\cdot)\) \(\chi_{2736}(1705,\cdot)\)

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: 18.0.207745302509684413663639240971181728202752.1

Values on generators

\((1711,2053,1217,1009)\) → \((1,-1,e\left(\frac{1}{3}\right),e\left(\frac{1}{18}\right))\)

Values

\(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(23\)\(25\)\(29\)\(31\)\(35\)
\(-1\)\(1\)\(e\left(\frac{1}{18}\right)\)\(e\left(\frac{2}{3}\right)\)\(-1\)\(e\left(\frac{4}{9}\right)\)\(e\left(\frac{5}{9}\right)\)\(e\left(\frac{7}{9}\right)\)\(e\left(\frac{1}{9}\right)\)\(e\left(\frac{7}{9}\right)\)\(-1\)\(e\left(\frac{13}{18}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2736 }(553,a) \;\) at \(\;a = \) e.g. 2