Properties

Label 1005.89
Modulus $1005$
Conductor $1005$
Order $22$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1005, base_ring=CyclotomicField(22))
 
M = H._module
 
chi = DirichletCharacter(H, M([11,11,20]))
 
pari: [g,chi] = znchar(Mod(89,1005))
 

Basic properties

Modulus: \(1005\)
Conductor: \(1005\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(22\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1005.bb

\(\chi_{1005}(14,\cdot)\) \(\chi_{1005}(59,\cdot)\) \(\chi_{1005}(89,\cdot)\) \(\chi_{1005}(149,\cdot)\) \(\chi_{1005}(344,\cdot)\) \(\chi_{1005}(359,\cdot)\) \(\chi_{1005}(464,\cdot)\) \(\chi_{1005}(494,\cdot)\) \(\chi_{1005}(509,\cdot)\) \(\chi_{1005}(734,\cdot)\)

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

\((671,202,136)\) → \((-1,-1,e\left(\frac{10}{11}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(11\)\(13\)\(14\)\(16\)\(17\)\(19\)
\( \chi_{ 1005 }(89, a) \) \(-1\)\(1\)\(e\left(\frac{10}{11}\right)\)\(e\left(\frac{9}{11}\right)\)\(e\left(\frac{9}{22}\right)\)\(e\left(\frac{8}{11}\right)\)\(e\left(\frac{3}{22}\right)\)\(e\left(\frac{17}{22}\right)\)\(e\left(\frac{7}{22}\right)\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{2}{11}\right)\)\(e\left(\frac{1}{11}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1005 }(89,a) \;\) at \(\;a = \) e.g. 2