Properties

Label 1089.208
Modulus $1089$
Conductor $121$
Order $22$
Real no
Primitive no
Minimal yes
Parity odd

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1089, base_ring=CyclotomicField(22))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,21]))
 
pari: [g,chi] = znchar(Mod(208,1089))
 

Basic properties

Modulus: \(1089\)
Conductor: \(121\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(22\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{121}(87,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1089.p

\(\chi_{1089}(10,\cdot)\) \(\chi_{1089}(109,\cdot)\) \(\chi_{1089}(208,\cdot)\) \(\chi_{1089}(307,\cdot)\) \(\chi_{1089}(406,\cdot)\) \(\chi_{1089}(505,\cdot)\) \(\chi_{1089}(703,\cdot)\) \(\chi_{1089}(802,\cdot)\) \(\chi_{1089}(901,\cdot)\) \(\chi_{1089}(1000,\cdot)\)

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: \(\Q(\zeta_{11})\)
Fixed field: Number field defined by a degree 22 polynomial

Values on generators

\((848,244)\) → \((1,e\left(\frac{21}{22}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(13\)\(14\)\(16\)\(17\)
\( \chi_{ 1089 }(208, a) \) \(-1\)\(1\)\(e\left(\frac{21}{22}\right)\)\(e\left(\frac{10}{11}\right)\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{15}{22}\right)\)\(e\left(\frac{19}{22}\right)\)\(e\left(\frac{13}{22}\right)\)\(e\left(\frac{9}{22}\right)\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{9}{11}\right)\)\(e\left(\frac{17}{22}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1089 }(208,a) \;\) at \(\;a = \) e.g. 2