Properties

Label 1296.17
Modulus $1296$
Conductor $27$
Order $18$
Real no
Primitive no
Minimal no
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1296, base_ring=CyclotomicField(18))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,0,11]))
 
pari: [g,chi] = znchar(Mod(17,1296))
 

Basic properties

Modulus: \(1296\)
Conductor: \(27\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(18\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{27}(23,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1296.bc

\(\chi_{1296}(17,\cdot)\) \(\chi_{1296}(305,\cdot)\) \(\chi_{1296}(449,\cdot)\) \(\chi_{1296}(737,\cdot)\) \(\chi_{1296}(881,\cdot)\) \(\chi_{1296}(1169,\cdot)\)

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

\((1135,325,1217)\) → \((1,1,e\left(\frac{11}{18}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 1296 }(17, a) \) \(-1\)\(1\)\(e\left(\frac{1}{18}\right)\)\(e\left(\frac{7}{9}\right)\)\(e\left(\frac{17}{18}\right)\)\(e\left(\frac{8}{9}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{13}{18}\right)\)\(e\left(\frac{1}{9}\right)\)\(e\left(\frac{11}{18}\right)\)\(e\left(\frac{2}{9}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1296 }(17,a) \;\) at \(\;a = \) e.g. 2