Properties

Label 1805.1689
Modulus $1805$
Conductor $95$
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(1805, base_ring=CyclotomicField(18))
 
M = H._module
 
chi = DirichletCharacter(H, M([9,10]))
 
pari: [g,chi] = znchar(Mod(1689,1805))
 

Basic properties

Modulus: \(1805\)
Conductor: \(95\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(18\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{95}(74,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1805.p

\(\chi_{1805}(54,\cdot)\) \(\chi_{1805}(99,\cdot)\) \(\chi_{1805}(234,\cdot)\) \(\chi_{1805}(389,\cdot)\) \(\chi_{1805}(784,\cdot)\) \(\chi_{1805}(1689,\cdot)\)

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: 18.18.563362135874260093126953125.1

Values on generators

\((362,1446)\) → \((-1,e\left(\frac{5}{9}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 1805 }(1689, a) \) \(1\)\(1\)\(e\left(\frac{1}{18}\right)\)\(e\left(\frac{13}{18}\right)\)\(e\left(\frac{1}{9}\right)\)\(e\left(\frac{7}{9}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{4}{9}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{5}{18}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1805 }(1689,a) \;\) at \(\;a = \) e.g. 2