Properties

Label 1000.987
Modulus $1000$
Conductor $1000$
Order $100$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1000, base_ring=CyclotomicField(100))
 
M = H._module
 
chi = DirichletCharacter(H, M([50,50,89]))
 
pari: [g,chi] = znchar(Mod(987,1000))
 

Basic properties

Modulus: \(1000\)
Conductor: \(1000\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(100\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1000.bh

\(\chi_{1000}(3,\cdot)\) \(\chi_{1000}(27,\cdot)\) \(\chi_{1000}(67,\cdot)\) \(\chi_{1000}(83,\cdot)\) \(\chi_{1000}(123,\cdot)\) \(\chi_{1000}(147,\cdot)\) \(\chi_{1000}(163,\cdot)\) \(\chi_{1000}(187,\cdot)\) \(\chi_{1000}(203,\cdot)\) \(\chi_{1000}(227,\cdot)\) \(\chi_{1000}(267,\cdot)\) \(\chi_{1000}(283,\cdot)\) \(\chi_{1000}(323,\cdot)\) \(\chi_{1000}(347,\cdot)\) \(\chi_{1000}(363,\cdot)\) \(\chi_{1000}(387,\cdot)\) \(\chi_{1000}(403,\cdot)\) \(\chi_{1000}(427,\cdot)\) \(\chi_{1000}(467,\cdot)\) \(\chi_{1000}(483,\cdot)\) \(\chi_{1000}(523,\cdot)\) \(\chi_{1000}(547,\cdot)\) \(\chi_{1000}(563,\cdot)\) \(\chi_{1000}(587,\cdot)\) \(\chi_{1000}(603,\cdot)\) \(\chi_{1000}(627,\cdot)\) \(\chi_{1000}(667,\cdot)\) \(\chi_{1000}(683,\cdot)\) \(\chi_{1000}(723,\cdot)\) \(\chi_{1000}(747,\cdot)\) ...

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: $\Q(\zeta_{100})$
Fixed field: Number field defined by a degree 100 polynomial

Values on generators

\((751,501,377)\) → \((-1,-1,e\left(\frac{89}{100}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 1000 }(987, a) \) \(1\)\(1\)\(e\left(\frac{23}{100}\right)\)\(e\left(\frac{3}{20}\right)\)\(e\left(\frac{23}{50}\right)\)\(e\left(\frac{16}{25}\right)\)\(e\left(\frac{21}{100}\right)\)\(e\left(\frac{97}{100}\right)\)\(e\left(\frac{1}{50}\right)\)\(e\left(\frac{19}{50}\right)\)\(e\left(\frac{9}{100}\right)\)\(e\left(\frac{69}{100}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1000 }(987,a) \;\) at \(\;a = \) e.g. 2