Properties

Label 1000.637
Modulus $1000$
Conductor $1000$
Order $100$
Real no
Primitive yes
Minimal yes
Parity odd

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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([0,50,9]))
 
pari: [g,chi] = znchar(Mod(637,1000))
 

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: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1000.bj

\(\chi_{1000}(13,\cdot)\) \(\chi_{1000}(37,\cdot)\) \(\chi_{1000}(53,\cdot)\) \(\chi_{1000}(77,\cdot)\) \(\chi_{1000}(117,\cdot)\) \(\chi_{1000}(133,\cdot)\) \(\chi_{1000}(173,\cdot)\) \(\chi_{1000}(197,\cdot)\) \(\chi_{1000}(213,\cdot)\) \(\chi_{1000}(237,\cdot)\) \(\chi_{1000}(253,\cdot)\) \(\chi_{1000}(277,\cdot)\) \(\chi_{1000}(317,\cdot)\) \(\chi_{1000}(333,\cdot)\) \(\chi_{1000}(373,\cdot)\) \(\chi_{1000}(397,\cdot)\) \(\chi_{1000}(413,\cdot)\) \(\chi_{1000}(437,\cdot)\) \(\chi_{1000}(453,\cdot)\) \(\chi_{1000}(477,\cdot)\) \(\chi_{1000}(517,\cdot)\) \(\chi_{1000}(533,\cdot)\) \(\chi_{1000}(573,\cdot)\) \(\chi_{1000}(597,\cdot)\) \(\chi_{1000}(613,\cdot)\) \(\chi_{1000}(637,\cdot)\) \(\chi_{1000}(653,\cdot)\) \(\chi_{1000}(677,\cdot)\) \(\chi_{1000}(717,\cdot)\) \(\chi_{1000}(733,\cdot)\) ...

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

\((751,501,377)\) → \((1,-1,e\left(\frac{9}{100}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 1000 }(637, a) \) \(-1\)\(1\)\(e\left(\frac{13}{100}\right)\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{13}{50}\right)\)\(e\left(\frac{17}{50}\right)\)\(e\left(\frac{1}{100}\right)\)\(e\left(\frac{57}{100}\right)\)\(e\left(\frac{3}{25}\right)\)\(e\left(\frac{39}{50}\right)\)\(e\left(\frac{79}{100}\right)\)\(e\left(\frac{39}{100}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1000 }(637,a) \;\) at \(\;a = \) e.g. 2