Properties

Label 1003.28
Modulus $1003$
Conductor $1003$
Order $464$
Real no
Primitive yes
Minimal yes
Parity odd

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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([203,160]))
 
pari: [g,chi] = znchar(Mod(28,1003))
 

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

Galois orbit 1003.s

\(\chi_{1003}(3,\cdot)\) \(\chi_{1003}(5,\cdot)\) \(\chi_{1003}(7,\cdot)\) \(\chi_{1003}(12,\cdot)\) \(\chi_{1003}(20,\cdot)\) \(\chi_{1003}(22,\cdot)\) \(\chi_{1003}(27,\cdot)\) \(\chi_{1003}(28,\cdot)\) \(\chi_{1003}(29,\cdot)\) \(\chi_{1003}(41,\cdot)\) \(\chi_{1003}(45,\cdot)\) \(\chi_{1003}(46,\cdot)\) \(\chi_{1003}(48,\cdot)\) \(\chi_{1003}(57,\cdot)\) \(\chi_{1003}(62,\cdot)\) \(\chi_{1003}(63,\cdot)\) \(\chi_{1003}(71,\cdot)\) \(\chi_{1003}(74,\cdot)\) \(\chi_{1003}(75,\cdot)\) \(\chi_{1003}(78,\cdot)\) \(\chi_{1003}(79,\cdot)\) \(\chi_{1003}(80,\cdot)\) \(\chi_{1003}(88,\cdot)\) \(\chi_{1003}(95,\cdot)\) \(\chi_{1003}(105,\cdot)\) \(\chi_{1003}(107,\cdot)\) \(\chi_{1003}(108,\cdot)\) \(\chi_{1003}(112,\cdot)\) \(\chi_{1003}(116,\cdot)\) \(\chi_{1003}(122,\cdot)\) ...

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

\((768,120)\) → \((e\left(\frac{7}{16}\right),e\left(\frac{10}{29}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 1003 }(28, a) \) \(-1\)\(1\)\(e\left(\frac{109}{232}\right)\)\(e\left(\frac{315}{464}\right)\)\(e\left(\frac{109}{116}\right)\)\(e\left(\frac{119}{464}\right)\)\(e\left(\frac{69}{464}\right)\)\(e\left(\frac{9}{464}\right)\)\(e\left(\frac{95}{232}\right)\)\(e\left(\frac{83}{232}\right)\)\(e\left(\frac{337}{464}\right)\)\(e\left(\frac{317}{464}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1003 }(28,a) \;\) at \(\;a = \) e.g. 2