Properties

Label 1004.3
Modulus $1004$
Conductor $1004$
Order $250$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1004, base_ring=CyclotomicField(250))
 
M = H._module
 
chi = DirichletCharacter(H, M([125,16]))
 
pari: [g,chi] = znchar(Mod(3,1004))
 

Basic properties

Modulus: \(1004\)
Conductor: \(1004\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(250\)
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 1004.p

\(\chi_{1004}(3,\cdot)\) \(\chi_{1004}(7,\cdot)\) \(\chi_{1004}(15,\cdot)\) \(\chi_{1004}(23,\cdot)\) \(\chi_{1004}(27,\cdot)\) \(\chi_{1004}(31,\cdot)\) \(\chi_{1004}(35,\cdot)\) \(\chi_{1004}(39,\cdot)\) \(\chi_{1004}(67,\cdot)\) \(\chi_{1004}(75,\cdot)\) \(\chi_{1004}(79,\cdot)\) \(\chi_{1004}(83,\cdot)\) \(\chi_{1004}(103,\cdot)\) \(\chi_{1004}(115,\cdot)\) \(\chi_{1004}(119,\cdot)\) \(\chi_{1004}(131,\cdot)\) \(\chi_{1004}(135,\cdot)\) \(\chi_{1004}(147,\cdot)\) \(\chi_{1004}(155,\cdot)\) \(\chi_{1004}(175,\cdot)\) \(\chi_{1004}(179,\cdot)\) \(\chi_{1004}(195,\cdot)\) \(\chi_{1004}(207,\cdot)\) \(\chi_{1004}(227,\cdot)\) \(\chi_{1004}(263,\cdot)\) \(\chi_{1004}(279,\cdot)\) \(\chi_{1004}(287,\cdot)\) \(\chi_{1004}(299,\cdot)\) \(\chi_{1004}(303,\cdot)\) \(\chi_{1004}(311,\cdot)\) ...

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: $\Q(\zeta_{125})$
Fixed field: Number field defined by a degree 250 polynomial (not computed)

Values on generators

\((503,257)\) → \((-1,e\left(\frac{8}{125}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 1004 }(3, a) \) \(-1\)\(1\)\(e\left(\frac{131}{250}\right)\)\(e\left(\frac{8}{25}\right)\)\(e\left(\frac{93}{250}\right)\)\(e\left(\frac{6}{125}\right)\)\(e\left(\frac{1}{250}\right)\)\(e\left(\frac{81}{125}\right)\)\(e\left(\frac{211}{250}\right)\)\(e\left(\frac{92}{125}\right)\)\(e\left(\frac{233}{250}\right)\)\(e\left(\frac{112}{125}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1004 }(3,a) \;\) at \(\;a = \) e.g. 2