Properties

Label 2683.73
Modulus $2683$
Conductor $2683$
Order $1341$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2683, base_ring=CyclotomicField(2682))
 
M = H._module
 
chi = DirichletCharacter(H, M([1952]))
 
pari: [g,chi] = znchar(Mod(73,2683))
 

Basic properties

Modulus: \(2683\)
Conductor: \(2683\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(1341\)
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 2683.k

\(\chi_{2683}(4,\cdot)\) \(\chi_{2683}(6,\cdot)\) \(\chi_{2683}(9,\cdot)\) \(\chi_{2683}(11,\cdot)\) \(\chi_{2683}(14,\cdot)\) \(\chi_{2683}(16,\cdot)\) \(\chi_{2683}(21,\cdot)\) \(\chi_{2683}(23,\cdot)\) \(\chi_{2683}(24,\cdot)\) \(\chi_{2683}(25,\cdot)\) \(\chi_{2683}(26,\cdot)\) \(\chi_{2683}(34,\cdot)\) \(\chi_{2683}(35,\cdot)\) \(\chi_{2683}(36,\cdot)\) \(\chi_{2683}(38,\cdot)\) \(\chi_{2683}(39,\cdot)\) \(\chi_{2683}(40,\cdot)\) \(\chi_{2683}(41,\cdot)\) \(\chi_{2683}(51,\cdot)\) \(\chi_{2683}(54,\cdot)\) \(\chi_{2683}(57,\cdot)\) \(\chi_{2683}(58,\cdot)\) \(\chi_{2683}(60,\cdot)\) \(\chi_{2683}(65,\cdot)\) \(\chi_{2683}(67,\cdot)\) \(\chi_{2683}(71,\cdot)\) \(\chi_{2683}(73,\cdot)\) \(\chi_{2683}(74,\cdot)\) \(\chi_{2683}(81,\cdot)\) \(\chi_{2683}(85,\cdot)\) ...

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: $\Q(\zeta_{1341})$
Fixed field: Number field defined by a degree 1341 polynomial (not computed)

Values on generators

\(2\) → \(e\left(\frac{976}{1341}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 2683 }(73, a) \) \(1\)\(1\)\(e\left(\frac{976}{1341}\right)\)\(e\left(\frac{910}{1341}\right)\)\(e\left(\frac{611}{1341}\right)\)\(e\left(\frac{116}{1341}\right)\)\(e\left(\frac{545}{1341}\right)\)\(e\left(\frac{140}{447}\right)\)\(e\left(\frac{82}{447}\right)\)\(e\left(\frac{479}{1341}\right)\)\(e\left(\frac{364}{447}\right)\)\(e\left(\frac{565}{1341}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2683 }(73,a) \;\) at \(\;a = \) e.g. 2