Properties

Label 6498.1111
Modulus $6498$
Conductor $171$
Order $9$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6498, base_ring=CyclotomicField(18))
 
M = H._module
 
chi = DirichletCharacter(H, M([6,8]))
 
pari: [g,chi] = znchar(Mod(1111,6498))
 

Basic properties

Modulus: \(6498\)
Conductor: \(171\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(9\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{171}(85,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6498.w

\(\chi_{6498}(1111,\cdot)\) \(\chi_{6498}(1543,\cdot)\) \(\chi_{6498}(2581,\cdot)\) \(\chi_{6498}(4927,\cdot)\) \(\chi_{6498}(5299,\cdot)\) \(\chi_{6498}(6199,\cdot)\)

sage: chi.galois_orbit()
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: \(\Q(\zeta_{9})\)
Fixed field: 9.9.9025761726072081.2

Values on generators

\((3611,6139)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{4}{9}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(23\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 6498 }(1111, a) \) \(1\)\(1\)\(e\left(\frac{7}{9}\right)\)\(1\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{8}{9}\right)\)\(e\left(\frac{4}{9}\right)\)\(e\left(\frac{5}{9}\right)\)\(e\left(\frac{5}{9}\right)\)\(e\left(\frac{8}{9}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{7}{9}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6498 }(1111,a) \;\) at \(\;a = \) e.g. 2