Properties

Label 1002.989
Modulus $1002$
Conductor $501$
Order $166$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1002, base_ring=CyclotomicField(166))
 
M = H._module
 
chi = DirichletCharacter(H, M([83,20]))
 
pari: [g,chi] = znchar(Mod(989,1002))
 

Basic properties

Modulus: \(1002\)
Conductor: \(501\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(166\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{501}(488,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1002.h

\(\chi_{1002}(11,\cdot)\) \(\chi_{1002}(29,\cdot)\) \(\chi_{1002}(47,\cdot)\) \(\chi_{1002}(65,\cdot)\) \(\chi_{1002}(77,\cdot)\) \(\chi_{1002}(89,\cdot)\) \(\chi_{1002}(107,\cdot)\) \(\chi_{1002}(137,\cdot)\) \(\chi_{1002}(173,\cdot)\) \(\chi_{1002}(179,\cdot)\) \(\chi_{1002}(185,\cdot)\) \(\chi_{1002}(191,\cdot)\) \(\chi_{1002}(203,\cdot)\) \(\chi_{1002}(209,\cdot)\) \(\chi_{1002}(215,\cdot)\) \(\chi_{1002}(221,\cdot)\) \(\chi_{1002}(233,\cdot)\) \(\chi_{1002}(239,\cdot)\) \(\chi_{1002}(251,\cdot)\) \(\chi_{1002}(263,\cdot)\) \(\chi_{1002}(275,\cdot)\) \(\chi_{1002}(281,\cdot)\) \(\chi_{1002}(293,\cdot)\) \(\chi_{1002}(299,\cdot)\) \(\chi_{1002}(311,\cdot)\) \(\chi_{1002}(317,\cdot)\) \(\chi_{1002}(329,\cdot)\) \(\chi_{1002}(341,\cdot)\) \(\chi_{1002}(353,\cdot)\) \(\chi_{1002}(359,\cdot)\) ...

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: $\Q(\zeta_{83})$
Fixed field: Number field defined by a degree 166 polynomial (not computed)

Values on generators

\((335,673)\) → \((-1,e\left(\frac{10}{83}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 1002 }(989, a) \) \(-1\)\(1\)\(e\left(\frac{103}{166}\right)\)\(e\left(\frac{18}{83}\right)\)\(e\left(\frac{145}{166}\right)\)\(e\left(\frac{34}{83}\right)\)\(e\left(\frac{147}{166}\right)\)\(e\left(\frac{82}{83}\right)\)\(e\left(\frac{71}{166}\right)\)\(e\left(\frac{20}{83}\right)\)\(e\left(\frac{95}{166}\right)\)\(e\left(\frac{70}{83}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1002 }(989,a) \;\) at \(\;a = \) e.g. 2