Properties

Label 1110.1061
Modulus $1110$
Conductor $111$
Order $18$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1110, base_ring=CyclotomicField(18))
 
M = H._module
 
chi = DirichletCharacter(H, M([9,0,5]))
 
pari: [g,chi] = znchar(Mod(1061,1110))
 

Basic properties

Modulus: \(1110\)
Conductor: \(111\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(18\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{111}(62,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1110.bs

\(\chi_{1110}(41,\cdot)\) \(\chi_{1110}(521,\cdot)\) \(\chi_{1110}(731,\cdot)\) \(\chi_{1110}(761,\cdot)\) \(\chi_{1110}(881,\cdot)\) \(\chi_{1110}(1061,\cdot)\)

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: 18.0.8985052139278849963819823767311.1

Values on generators

\((371,667,631)\) → \((-1,1,e\left(\frac{5}{18}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(41\)\(43\)
\( \chi_{ 1110 }(1061, a) \) \(-1\)\(1\)\(e\left(\frac{8}{9}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{1}{18}\right)\)\(e\left(\frac{4}{9}\right)\)\(e\left(\frac{13}{18}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{3}\right)\)\(-1\)\(e\left(\frac{1}{18}\right)\)\(-1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1110 }(1061,a) \;\) at \(\;a = \) e.g. 2