Properties

Label 1053.64
Modulus $1053$
Conductor $351$
Order $18$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1053, base_ring=CyclotomicField(18))
 
M = H._module
 
chi = DirichletCharacter(H, M([2,9]))
 
pari: [g,chi] = znchar(Mod(64,1053))
 

Basic properties

Modulus: \(1053\)
Conductor: \(351\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(18\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{351}(220,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1053.bl

\(\chi_{1053}(64,\cdot)\) \(\chi_{1053}(181,\cdot)\) \(\chi_{1053}(415,\cdot)\) \(\chi_{1053}(532,\cdot)\) \(\chi_{1053}(766,\cdot)\) \(\chi_{1053}(883,\cdot)\)

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: Number field defined by a degree 18 polynomial

Values on generators

\((326,730)\) → \((e\left(\frac{1}{9}\right),-1)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(14\)\(16\)\(17\)
\( \chi_{ 1053 }(64, a) \) \(1\)\(1\)\(e\left(\frac{11}{18}\right)\)\(e\left(\frac{2}{9}\right)\)\(e\left(\frac{1}{18}\right)\)\(e\left(\frac{5}{18}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{17}{18}\right)\)\(e\left(\frac{8}{9}\right)\)\(e\left(\frac{4}{9}\right)\)\(e\left(\frac{2}{3}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1053 }(64,a) \;\) at \(\;a = \) e.g. 2