Properties

Label 1003.24
Modulus $1003$
Conductor $1003$
Order $464$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1003, base_ring=CyclotomicField(464))
 
M = H._module
 
chi = DirichletCharacter(H, M([319,424]))
 
pari: [g,chi] = znchar(Mod(24,1003))
 

Basic properties

Modulus: \(1003\)
Conductor: \(1003\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(464\)
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 1003.t

\(\chi_{1003}(6,\cdot)\) \(\chi_{1003}(10,\cdot)\) \(\chi_{1003}(11,\cdot)\) \(\chi_{1003}(14,\cdot)\) \(\chi_{1003}(23,\cdot)\) \(\chi_{1003}(24,\cdot)\) \(\chi_{1003}(31,\cdot)\) \(\chi_{1003}(37,\cdot)\) \(\chi_{1003}(39,\cdot)\) \(\chi_{1003}(40,\cdot)\) \(\chi_{1003}(44,\cdot)\) \(\chi_{1003}(54,\cdot)\) \(\chi_{1003}(56,\cdot)\) \(\chi_{1003}(61,\cdot)\) \(\chi_{1003}(65,\cdot)\) \(\chi_{1003}(73,\cdot)\) \(\chi_{1003}(82,\cdot)\) \(\chi_{1003}(90,\cdot)\) \(\chi_{1003}(91,\cdot)\) \(\chi_{1003}(92,\cdot)\) \(\chi_{1003}(96,\cdot)\) \(\chi_{1003}(97,\cdot)\) \(\chi_{1003}(99,\cdot)\) \(\chi_{1003}(109,\cdot)\) \(\chi_{1003}(113,\cdot)\) \(\chi_{1003}(114,\cdot)\) \(\chi_{1003}(124,\cdot)\) \(\chi_{1003}(126,\cdot)\) \(\chi_{1003}(129,\cdot)\) \(\chi_{1003}(131,\cdot)\) ...

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: $\Q(\zeta_{464})$
Fixed field: Number field defined by a degree 464 polynomial (not computed)

Values on generators

\((768,120)\) → \((e\left(\frac{11}{16}\right),e\left(\frac{53}{58}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 1003 }(24, a) \) \(1\)\(1\)\(e\left(\frac{125}{232}\right)\)\(e\left(\frac{175}{464}\right)\)\(e\left(\frac{9}{116}\right)\)\(e\left(\frac{427}{464}\right)\)\(e\left(\frac{425}{464}\right)\)\(e\left(\frac{5}{464}\right)\)\(e\left(\frac{143}{232}\right)\)\(e\left(\frac{175}{232}\right)\)\(e\left(\frac{213}{464}\right)\)\(e\left(\frac{305}{464}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1003 }(24,a) \;\) at \(\;a = \) e.g. 2