Properties

Label 1089.19
Modulus $1089$
Conductor $121$
Order $110$
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(110))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,83]))
 
pari: [g,chi] = znchar(Mod(19,1089))
 

Basic properties

Modulus: \(1089\)
Conductor: \(121\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(110\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{121}(19,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1089.ba

\(\chi_{1089}(19,\cdot)\) \(\chi_{1089}(28,\cdot)\) \(\chi_{1089}(46,\cdot)\) \(\chi_{1089}(73,\cdot)\) \(\chi_{1089}(127,\cdot)\) \(\chi_{1089}(145,\cdot)\) \(\chi_{1089}(172,\cdot)\) \(\chi_{1089}(217,\cdot)\) \(\chi_{1089}(226,\cdot)\) \(\chi_{1089}(244,\cdot)\) \(\chi_{1089}(271,\cdot)\) \(\chi_{1089}(316,\cdot)\) \(\chi_{1089}(325,\cdot)\) \(\chi_{1089}(343,\cdot)\) \(\chi_{1089}(370,\cdot)\) \(\chi_{1089}(415,\cdot)\) \(\chi_{1089}(424,\cdot)\) \(\chi_{1089}(442,\cdot)\) \(\chi_{1089}(469,\cdot)\) \(\chi_{1089}(514,\cdot)\) \(\chi_{1089}(523,\cdot)\) \(\chi_{1089}(541,\cdot)\) \(\chi_{1089}(568,\cdot)\) \(\chi_{1089}(613,\cdot)\) \(\chi_{1089}(622,\cdot)\) \(\chi_{1089}(640,\cdot)\) \(\chi_{1089}(667,\cdot)\) \(\chi_{1089}(712,\cdot)\) \(\chi_{1089}(721,\cdot)\) \(\chi_{1089}(739,\cdot)\) ...

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: $\Q(\zeta_{55})$
Fixed field: Number field defined by a degree 110 polynomial (not computed)

Values on generators

\((848,244)\) → \((1,e\left(\frac{83}{110}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(13\)\(14\)\(16\)\(17\)
\( \chi_{ 1089 }(19, a) \) \(-1\)\(1\)\(e\left(\frac{83}{110}\right)\)\(e\left(\frac{28}{55}\right)\)\(e\left(\frac{46}{55}\right)\)\(e\left(\frac{31}{110}\right)\)\(e\left(\frac{29}{110}\right)\)\(e\left(\frac{13}{22}\right)\)\(e\left(\frac{23}{110}\right)\)\(e\left(\frac{2}{55}\right)\)\(e\left(\frac{1}{55}\right)\)\(e\left(\frac{107}{110}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1089 }(19,a) \;\) at \(\;a = \) e.g. 2