Properties

Label 5400.851
Modulus $5400$
Conductor $216$
Order $18$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5400, base_ring=CyclotomicField(18))
 
M = H._module
 
chi = DirichletCharacter(H, M([9,9,17,0]))
 
pari: [g,chi] = znchar(Mod(851,5400))
 

Basic properties

Modulus: \(5400\)
Conductor: \(216\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(18\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{216}(203,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 5400.cy

\(\chi_{5400}(851,\cdot)\) \(\chi_{5400}(1451,\cdot)\) \(\chi_{5400}(2651,\cdot)\) \(\chi_{5400}(3251,\cdot)\) \(\chi_{5400}(4451,\cdot)\) \(\chi_{5400}(5051,\cdot)\)

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: 18.18.396521139274783615537700143104.1

Values on generators

\((1351,2701,1001,2377)\) → \((-1,-1,e\left(\frac{17}{18}\right),1)\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 5400 }(851, a) \) \(1\)\(1\)\(e\left(\frac{11}{18}\right)\)\(e\left(\frac{5}{18}\right)\)\(e\left(\frac{1}{18}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{8}{9}\right)\)\(e\left(\frac{4}{9}\right)\)\(e\left(\frac{7}{18}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{1}{18}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 5400 }(851,a) \;\) at \(\;a = \) e.g. 2