Properties

Label 1279.3
Modulus $1279$
Conductor $1279$
Order $1278$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1279, base_ring=CyclotomicField(1278))
 
M = H._module
 
chi = DirichletCharacter(H, M([1]))
 
pari: [g,chi] = znchar(Mod(3,1279))
 

Basic properties

Modulus: \(1279\)
Conductor: \(1279\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(1278\)
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 1279.l

\(\chi_{1279}(3,\cdot)\) \(\chi_{1279}(6,\cdot)\) \(\chi_{1279}(13,\cdot)\) \(\chi_{1279}(15,\cdot)\) \(\chi_{1279}(21,\cdot)\) \(\chi_{1279}(22,\cdot)\) \(\chi_{1279}(24,\cdot)\) \(\chi_{1279}(26,\cdot)\) \(\chi_{1279}(29,\cdot)\) \(\chi_{1279}(31,\cdot)\) \(\chi_{1279}(38,\cdot)\) \(\chi_{1279}(44,\cdot)\) \(\chi_{1279}(48,\cdot)\) \(\chi_{1279}(54,\cdot)\) \(\chi_{1279}(55,\cdot)\) \(\chi_{1279}(60,\cdot)\) \(\chi_{1279}(61,\cdot)\) \(\chi_{1279}(62,\cdot)\) \(\chi_{1279}(65,\cdot)\) \(\chi_{1279}(67,\cdot)\) \(\chi_{1279}(69,\cdot)\) \(\chi_{1279}(76,\cdot)\) \(\chi_{1279}(77,\cdot)\) \(\chi_{1279}(84,\cdot)\) \(\chi_{1279}(89,\cdot)\) \(\chi_{1279}(91,\cdot)\) \(\chi_{1279}(95,\cdot)\) \(\chi_{1279}(99,\cdot)\) \(\chi_{1279}(101,\cdot)\) \(\chi_{1279}(102,\cdot)\) ...

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: $\Q(\zeta_{639})$
Fixed field: Number field defined by a degree 1278 polynomial (not computed)

Values on generators

\(3\) → \(e\left(\frac{1}{1278}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 1279 }(3, a) \) \(-1\)\(1\)\(e\left(\frac{23}{639}\right)\)\(e\left(\frac{1}{1278}\right)\)\(e\left(\frac{46}{639}\right)\)\(e\left(\frac{455}{639}\right)\)\(e\left(\frac{47}{1278}\right)\)\(e\left(\frac{356}{639}\right)\)\(e\left(\frac{23}{213}\right)\)\(e\left(\frac{1}{639}\right)\)\(e\left(\frac{478}{639}\right)\)\(e\left(\frac{19}{426}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1279 }(3,a) \;\) at \(\;a = \) e.g. 2