Properties

Label 2000.571
Modulus $2000$
Conductor $2000$
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(2000, base_ring=CyclotomicField(100))
 
M = H._module
 
chi = DirichletCharacter(H, M([50,25,72]))
 
pari: [g,chi] = znchar(Mod(571,2000))
 

Basic properties

Modulus: \(2000\)
Conductor: \(2000\)
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 2000.bw

\(\chi_{2000}(11,\cdot)\) \(\chi_{2000}(91,\cdot)\) \(\chi_{2000}(131,\cdot)\) \(\chi_{2000}(171,\cdot)\) \(\chi_{2000}(211,\cdot)\) \(\chi_{2000}(291,\cdot)\) \(\chi_{2000}(331,\cdot)\) \(\chi_{2000}(371,\cdot)\) \(\chi_{2000}(411,\cdot)\) \(\chi_{2000}(491,\cdot)\) \(\chi_{2000}(531,\cdot)\) \(\chi_{2000}(571,\cdot)\) \(\chi_{2000}(611,\cdot)\) \(\chi_{2000}(691,\cdot)\) \(\chi_{2000}(731,\cdot)\) \(\chi_{2000}(771,\cdot)\) \(\chi_{2000}(811,\cdot)\) \(\chi_{2000}(891,\cdot)\) \(\chi_{2000}(931,\cdot)\) \(\chi_{2000}(971,\cdot)\) \(\chi_{2000}(1011,\cdot)\) \(\chi_{2000}(1091,\cdot)\) \(\chi_{2000}(1131,\cdot)\) \(\chi_{2000}(1171,\cdot)\) \(\chi_{2000}(1211,\cdot)\) \(\chi_{2000}(1291,\cdot)\) \(\chi_{2000}(1331,\cdot)\) \(\chi_{2000}(1371,\cdot)\) \(\chi_{2000}(1411,\cdot)\) \(\chi_{2000}(1491,\cdot)\) ...

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

\((751,501,1377)\) → \((-1,i,e\left(\frac{18}{25}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 2000 }(571, a) \) \(-1\)\(1\)\(e\left(\frac{29}{100}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{29}{50}\right)\)\(e\left(\frac{47}{100}\right)\)\(e\left(\frac{83}{100}\right)\)\(e\left(\frac{14}{25}\right)\)\(e\left(\frac{21}{100}\right)\)\(e\left(\frac{49}{100}\right)\)\(e\left(\frac{8}{25}\right)\)\(e\left(\frac{87}{100}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2000 }(571,a) \;\) at \(\;a = \) e.g. 2