Properties

Label 1011.983
Modulus $1011$
Conductor $1011$
Order $168$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1011, base_ring=CyclotomicField(168))
 
M = H._module
 
chi = DirichletCharacter(H, M([84,43]))
 
pari: [g,chi] = znchar(Mod(983,1011))
 

Basic properties

Modulus: \(1011\)
Conductor: \(1011\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(168\)
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 1011.bk

\(\chi_{1011}(14,\cdot)\) \(\chi_{1011}(50,\cdot)\) \(\chi_{1011}(86,\cdot)\) \(\chi_{1011}(95,\cdot)\) \(\chi_{1011}(107,\cdot)\) \(\chi_{1011}(113,\cdot)\) \(\chi_{1011}(149,\cdot)\) \(\chi_{1011}(155,\cdot)\) \(\chi_{1011}(167,\cdot)\) \(\chi_{1011}(170,\cdot)\) \(\chi_{1011}(182,\cdot)\) \(\chi_{1011}(188,\cdot)\) \(\chi_{1011}(224,\cdot)\) \(\chi_{1011}(230,\cdot)\) \(\chi_{1011}(242,\cdot)\) \(\chi_{1011}(251,\cdot)\) \(\chi_{1011}(287,\cdot)\) \(\chi_{1011}(323,\cdot)\) \(\chi_{1011}(365,\cdot)\) \(\chi_{1011}(428,\cdot)\) \(\chi_{1011}(431,\cdot)\) \(\chi_{1011}(437,\cdot)\) \(\chi_{1011}(449,\cdot)\) \(\chi_{1011}(452,\cdot)\) \(\chi_{1011}(482,\cdot)\) \(\chi_{1011}(527,\cdot)\) \(\chi_{1011}(548,\cdot)\) \(\chi_{1011}(566,\cdot)\) \(\chi_{1011}(596,\cdot)\) \(\chi_{1011}(611,\cdot)\) ...

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: $\Q(\zeta_{168})$
Fixed field: Number field defined by a degree 168 polynomial (not computed)

Values on generators

\((338,10)\) → \((-1,e\left(\frac{43}{168}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 1011 }(983, a) \) \(-1\)\(1\)\(e\left(\frac{13}{42}\right)\)\(e\left(\frac{13}{21}\right)\)\(e\left(\frac{53}{56}\right)\)\(e\left(\frac{11}{28}\right)\)\(e\left(\frac{13}{14}\right)\)\(e\left(\frac{43}{168}\right)\)\(e\left(\frac{15}{56}\right)\)\(e\left(\frac{2}{21}\right)\)\(e\left(\frac{59}{84}\right)\)\(e\left(\frac{5}{21}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1011 }(983,a) \;\) at \(\;a = \) e.g. 2